raw_data/tabular_datasets/c6/c6-val.csv

Student ID;Student Country;Question ID;Type of Answer;Question Level;Topic;Subtopic;Keywords
91;Lithuania;447;1;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
1340;Portugal;1118;0;Basic;Numerical Methods;Numerical Methods;Newton’s interpolation,Polynomial interpolation
576;Ireland;553;0;Basic;Analytic Geometry;Analytic Geometry;Orthogonality,Distance,Scalar product
1220;Slovenia;324;1;Basic;Linear Algebra;Linear Transformations;Linearity,Linear application
525;Portugal;612;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1220;Slovenia;323;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
899;Lithuania;423;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
511;Portugal;786;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
52;Portugal;141;1;Basic;Linear Algebra;Matrices and Determinants;Matrix operations,Matrix multiplication
898;Portugal;446;0;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
394;Portugal;345;1;Advanced;Complex Numbers;Complex Numbers;Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas
1071;Lithuania;438;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
37;Portugal;329;1;Basic;Complex Numbers;Complex Numbers;Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number
37;Portugal;546;0;Basic;Analytic Geometry;Analytic Geometry;Orthogonal projection,Distance,Cross product,Orthogonality
72;Portugal;439;0;Advanced;Linear Algebra;Vector Spaces;Span,Subspace,Linear combination
91;Lithuania;463;1;Basic;Fundamental Mathematics;Elementary Geometry;Polyhedrons,Geometric solids,Classification of geometric solids
690;Portugal;181;0;Basic;Statistics;Statistics;Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode
509;Portugal;383;0;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
1081;Portugal;447;1;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
411;Portugal;618;1;Advanced;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial
378;Portugal;277;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
168;Portugal;384;1;Basic;Linear Algebra;Linear Transformations;Linearity,Invertible linear operator,Composition of linear applications,Linear application
689;Portugal;863;1;Basic;Optimization;Nonlinear Optimization;Saddle point,Optimal solution,Minimum,Maximum,Stationary point
80;Lithuania;554;1;Basic;Analytic Geometry;Analytic Geometry;Scalar product,Orthogonality,Orthogonal projection,Distance
855;Italy;234;0;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
511;Portugal;221;0;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
1402;Portugal;289;1;Advanced;Linear Algebra;Linear Transformations;Kernel,Isomorphism,Injective linear application,Range
1286;Italy;416;1;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
462;Lithuania;647;1;Basic;Linear Algebra;Matrices and Determinants;Matrix multiplication,Symmetric matrix,Matrix operations
1256;Portugal;443;1;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism
168;Portugal;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1299;Italy;431;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
168;Portugal;292;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism
473;Portugal;383;0;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
1220;Slovenia;444;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
375;Portugal;317;0;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
1146;Portugal;455;0;Basic;Linear Algebra;Vector Spaces;Basis
37;Portugal;1100;1;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Newton’s interpolation
979;Portugal;1113;1;Basic;Numerical Methods;Numerical Methods;Newton’s interpolation,Polynomial interpolation
1023;Lithuania;426;0;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
1028;Lithuania;875;0;Basic;Differentiation;Partial Differentiation;Gradient,First order
913;Ireland;375;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1156;Portugal;421;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
955;Lithuania;1092;0;Basic;Numerical Methods;Numerical Methods;Newton’s method,Nonlinear equation
181;Italy;923;0;Basic;Real Functions of a single variable;Domain, Image and Graphics;Domain
955;Lithuania;451;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
525;Portugal;102;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
695;Portugal;211;0;Basic;Statistics;Statistics;Linear regression
535;Portugal;313;0;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
1140;Portugal;321;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
986;Portugal;783;1;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Isomorphism
72;Portugal;327;0;Basic;Complex Numbers;Complex Numbers;Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number
1219;Portugal;376;0;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1340;Portugal;1108;0;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Lagrange's interpolation
1147;Portugal;422;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
1080;Portugal;778;0;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1363;Portugal;427;1;Advanced;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
411;Portugal;160;0;Basic;Differential Equations;Differential Equations;Linear differential equation
634;Italy;776;1;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
551;Lithuania;701;1;Basic;Analytic Geometry;Analytic Geometry;Orthogonality,Cross product,Distance
468;Portugal;357;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers
464;Portugal;357;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers
1428;Portugal;383;1;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
913;Ireland;1104;1;Basic;Numerical Methods;Numerical Methods;Newton’s method,Nonlinear equation
1075;Portugal;257;0;Basic;Differentiation;Partial Differentiation;Gradient,First order
1402;Portugal;781;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
204;Lithuania;108;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1087;Portugal;426;0;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
624;Portugal;367;1;Basic;Fundamental Mathematics;Elementary Geometry;Geometrical figures
128;Lithuania;353;0;Basic;Complex Numbers;Complex Numbers;Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas
380;Portugal;355;0;Advanced;Complex Numbers;Complex Numbers;Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas
973;Portugal;435;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
1220;Slovenia;455;1;Basic;Linear Algebra;Vector Spaces;Basis
340;Portugal;303;0;Basic;Probability ;Probability ;Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability
175;Italy;143;0;Basic;Differentiation;Derivatives;Trigonometric rules,Logarithmic rule
511;Portugal;224;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
625;Russian Federation;112;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1517;Portugal;382;0;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
634;Italy;455;0;Basic;Linear Algebra;Vector Spaces;Basis
1339;Portugal;439;1;Advanced;Linear Algebra;Vector Spaces;Span,Subspace,Linear combination
986;Portugal;431;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
576;Ireland;550;1;Advanced;Analytic Geometry;Analytic Geometry;Cross product,Orthogonality
37;Portugal;904;0;Basic;Differential Equations;Differential Equations;Bernoulli equation,Linear differential equation
511;Portugal;108;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1037;Lithuania;264;0;Basic;Differentiation;Partial Differentiation;Chain rule
168;Portugal;439;0;Advanced;Linear Algebra;Vector Spaces;Span,Subspace,Linear combination
876;Italy;453;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1179;Portugal;459;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
982;Portugal;420;1;Advanced;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
421;Portugal;353;0;Basic;Complex Numbers;Complex Numbers;Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas
1219;Portugal;377;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel,Matrix of a linear transformation
72;Portugal;434;1;Basic;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
898;Portugal;438;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
26;Portugal;156;1;Basic;Differential Equations;Differential Equations;Linear differential equation
511;Portugal;134;1;Basic;Linear Algebra;Matrices and Determinants;Determinant computation,Determinant properties
1220;Slovenia;457;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
1028;Lithuania;999;0;Basic;Differentiation;Partial Differentiation;First order
840;Italy;319;1;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation,Linear application,Linearity
1353;Portugal;381;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
1564;Portugal;753;0;Basic;Differentiation;Derivatives;Exponential rule
468;Portugal;347;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
71;Portugal;642;1;Basic;Linear Algebra;Matrices and Determinants;Symmetric matrix,Matrix multiplication,Matrix operations
81;Lithuania;568;1;Basic;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Cartesian equations of a line
1179;Portugal;450;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
71;Portugal;118;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
966;Portugal;383;0;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
376;Portugal;347;0;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
167;Portugal;383;1;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
972;Portugal;441;0;Advanced;Linear Algebra;Vector Spaces;Basis,Span
189;Portugal;273;0;Basic;Differentiation;Derivatives;Product rule,Trigonometric rules
851;Italy;379;0;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
986;Portugal;448;1;Advanced;Linear Algebra;Vector Spaces;Subspace
1023;Lithuania;431;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
955;Lithuania;1088;0;Basic;Numerical Methods;Numerical Methods;First order,Second order
1220;Slovenia;325;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel
168;Portugal;324;0;Basic;Linear Algebra;Linear Transformations;Linearity,Linear application
1076;Portugal;437;0;Basic;Linear Algebra;Linear Transformations;Linear application,Composition of linear applications
128;Lithuania;334;0;Basic;Complex Numbers;Complex Numbers;Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form
37;Portugal;176;1;Advanced;Differentiation;Partial Differentiation;Second order,First order
972;Portugal;777;0;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
71;Portugal;330;0;Basic;Complex Numbers;Complex Numbers;Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number
840;Italy;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1322;Italy;423;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1135;Portugal;278;0;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
524;Portugal;496;0;Basic;Fundamental Mathematics;Elementary Geometry;Polygons
1023;Lithuania;431;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
913;Ireland;1252;1;Basic;Numerical Methods;Numerical Methods;Newton’s interpolation,Polynomial interpolation
61;Portugal;168;1;Basic;Differential Equations;Differential Equations;Linear differential equation
1220;Slovenia;379;1;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
1288;Portugal;450;0;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
664;Portugal;178;0;Basic;Statistics;Statistics;Mean,Variance,Standard deviation,Median,Mode
986;Portugal;777;1;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1505;Portugal;1003;0;Basic;Optimization;Linear Optimization;Linear programming
1276;Portugal;421;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
876;Italy;459;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
176;Italy;661;0;Basic;Differentiation;Derivatives;Logarithmic rule,Product rule,Trigonometric rules
393;Lithuania;356;1;Advanced;Complex Numbers;Complex Numbers;Operations with complex numbers
1109;Portugal;417;0;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
91;Lithuania;443;1;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism
890;Portugal;417;0;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
380;Portugal;1014;1;Basic;Optimization;Nonlinear Optimization;Optimal solution,Stationary point,Minimum,Maximum,Saddle point
152;Portugal;339;1;Basic;Complex Numbers;Complex Numbers;Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation
131;Lithuania;373;1;Advanced;Graph Theory;Graph Theory;Simple graph
405;Portugal;613;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1420;Portugal;382;1;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
1220;Slovenia;452;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1220;Slovenia;376;1;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation
71;Portugal;611;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1271;Portugal;382;1;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
1247;Portugal;356;0;Advanced;Complex Numbers;Complex Numbers;Operations with complex numbers
1340;Portugal;1087;1;Basic;Numerical Methods;Numerical Methods;First order,Second order
988;Portugal;223;0;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
1220;Slovenia;319;1;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation,Linear application,Linearity
72;Portugal;335;0;Basic;Complex Numbers;Complex Numbers;Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form
1087;Portugal;436;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
576;Ireland;565;1;Basic;Analytic Geometry;Analytic Geometry;Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus
913;Ireland;771;1;Basic;Probability ;Probability ;Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability
913;Ireland;1106;1;Basic;Numerical Methods;Numerical Methods;Newton’s method,Nonlinear equation
913;Ireland;1111;1;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Newton’s interpolation
466;Portugal;350;0;Advanced;Complex Numbers;Complex Numbers;Complex plane 
510;Portugal;323;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1324;Italy;287;0;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1220;Slovenia;435;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
1288;Portugal;453;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
151;Portugal;113;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1308;Italy;313;1;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
37;Portugal;1091;0;Basic;Numerical Methods;Numerical Methods;Nonlinear equation,Newton’s method
1220;Slovenia;423;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
81;Lithuania;480;1;Basic;Graph Theory;Graph Theory;Planar graph
1543;Portugal;1114;0;Basic;Numerical Methods;Numerical Methods;Newton’s interpolation,Polynomial interpolation
1023;Lithuania;421;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
457;Portugal;326;1;Basic;Complex Numbers;Complex Numbers;Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part
80;Lithuania;668;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
415;Portugal;328;0;Basic;Complex Numbers;Complex Numbers;Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number
1274;Portugal;118;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
509;Portugal;605;1;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial
1135;Portugal;289;0;Advanced;Linear Algebra;Linear Transformations;Kernel,Isomorphism,Injective linear application,Range
1220;Slovenia;457;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
625;Russian Federation;118;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1229;Portugal;290;0;Advanced;Linear Algebra;Linear Transformations;Linear application,Linearity
1284;Italy;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
151;Portugal;113;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
633;Italy;384;1;Basic;Linear Algebra;Linear Transformations;Linearity,Invertible linear operator,Composition of linear applications,Linear application
582;Lithuania;700;0;Basic;Analytic Geometry;Analytic Geometry;Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance
37;Portugal;773;0;Basic;Probability ;Probability ;Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability
91;Lithuania;448;1;Advanced;Linear Algebra;Vector Spaces;Subspace
90;Lithuania;139;0;Advanced;Linear Algebra;Matrices and Determinants;Matrix operations,Matrix multiplication,Determinant computation,Determinant properties
1543;Portugal;1113;0;Basic;Numerical Methods;Numerical Methods;Newton’s interpolation,Polynomial interpolation
511;Portugal;464;0;Basic;Fundamental Mathematics;Elementary Geometry;Polygons,Geometrical figures
1022;Lithuania;279;0;Basic;Optimization;Linear Optimization;Linear programming
1334;Portugal;434;0;Basic;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
1046;Lithuania;877;0;Basic;Differentiation;Partial Differentiation;Second order
1146;Portugal;443;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism
200;Lithuania;394;1;Basic;Integration;Integration Techniques;Substitution
483;Portugal;440;1;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Subspace,Linear combination,Dimension
1220;Slovenia;455;0;Basic;Linear Algebra;Vector Spaces;Basis
91;Lithuania;508;1;Advanced;Fundamental Mathematics;Elementary Geometry;Symmetry
1173;Portugal;277;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1147;Portugal;435;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
81;Lithuania;471;1;Advanced;Graph Theory;Graph Theory;Simple graph
175;Italy;339;0;Basic;Complex Numbers;Complex Numbers;Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation
1283;Italy;423;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
470;Portugal;347;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
344;Portugal;120;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
524;Portugal;231;1;Basic;Linear Algebra;Linear Systems;Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system
91;Lithuania;336;0;Basic;Complex Numbers;Complex Numbers;Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part
884;Italy;453;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1194;Portugal;785;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1220;Slovenia;428;1;Basic;Linear Algebra;Vector Spaces;Change-of-basis matrix,Span,Basis,Dimension
91;Lithuania;455;1;Basic;Linear Algebra;Vector Spaces;Basis
639;Italy;191;0;Basic;Statistics;Statistics;Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance
1272;Portugal;433;0;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
1220;Slovenia;436;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
1341;Portugal;450;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
1292;Italy;306;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism,Injective linear application
71;Portugal;342;1;Basic;Complex Numbers;Complex Numbers;Trigonometric form,Modulus of a complex number,Algebraic form
155;Spain;143;0;Basic;Differentiation;Derivatives;Trigonometric rules,Logarithmic rule
1147;Portugal;457;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
175;Italy;1087;1;Basic;Numerical Methods;Numerical Methods;First order,Second order
855;Italy;91;0;Basic;Linear Algebra;Linear Systems;Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system
1284;Italy;452;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1286;Italy;313;0;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
375;Portugal;358;0;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part
956;Portugal;375;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1196;Slovenia;434;1;Basic;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
198;Italy;118;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
411;Portugal;413;0;Basic;Integration;Integration Techniques;Direct inspection
41;Portugal;571;1;Basic;Analytic Geometry;Analytic Geometry;Intersection,Cartesian equations of a plane
898;Portugal;375;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
181;Italy;925;0;Basic;Real Functions of a single variable;Domain, Image and Graphics;Planar graph
1182;Portugal;425;1;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
525;Portugal;795;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Square roots,Simplify expressions
1253;Portugal;457;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
913;Ireland;419;1;Advanced;Linear Algebra;Vector Spaces;Subspace,Dimension
1220;Slovenia;423;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1023;Lithuania;422;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
407;Portugal;729;0;Basic;Differentiation;Derivatives;Quotient rule
36;Portugal;143;1;Basic;Differentiation;Derivatives;Trigonometric rules,Logarithmic rule
464;Portugal;352;1;Advanced;Complex Numbers;Complex Numbers;Conjugate number,Operations with complex numbers
1220;Slovenia;421;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
663;Portugal;180;1;Basic;Statistics;Statistics;Mean,Median,Mode
1023;Lithuania;438;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
955;Lithuania;1090;0;Basic;Numerical Methods;Numerical Methods;Nonlinear equation,Newton’s method
1350;Portugal;287;0;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1220;Slovenia;378;1;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
1286;Italy;426;0;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
1428;Portugal;306;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism,Injective linear application
582;Lithuania;347;0;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
898;Portugal;432;0;Advanced;Linear Algebra;Vector Spaces;Linear independence,Subspace,Linear combination,Linear dependence,Span
639;Italy;77;0;Basic;Statistics;Statistics;Stem and Leaf diagram,Relative frequency,Sample,Frequency
1093;Portugal;380;0;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1220;Slovenia;323;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
80;Lithuania;244;0;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
1289;Italy;448;0;Advanced;Linear Algebra;Vector Spaces;Subspace
913;Ireland;307;1;Basic;Probability ;Probability ;Event,Addition, multiplication, and total probability rules,Independence
1284;Italy;427;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
1023;Lithuania;421;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1109;Portugal;434;0;Basic;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
511;Portugal;91;1;Basic;Linear Algebra;Linear Systems;Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system
1272;Portugal;422;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
1339;Portugal;430;1;Advanced;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1220;Slovenia;417;1;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
311;Ireland;635;0;Basic;Linear Algebra;Matrices and Determinants;Determinant computation,Determinant properties,Inverse matrix ,Matrix operations
91;Lithuania;518;1;Basic;Fundamental Mathematics;Elementary Geometry;Polygons
668;Portugal;204;0;Basic;Statistics;Statistics;Student distribution,Hypothesis testing
1340;Portugal;1112;1;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Newton’s interpolation
1220;Slovenia;384;1;Basic;Linear Algebra;Linear Transformations;Linearity,Invertible linear operator,Composition of linear applications,Linear application
1220;Slovenia;433;1;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
634;Italy;457;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
1271;Portugal;435;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
634;Italy;378;1;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
1351;Portugal;306;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism,Injective linear application
311;Ireland;122;1;Basic;Linear Algebra;Matrices and Determinants;Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix
979;Portugal;1104;1;Basic;Numerical Methods;Numerical Methods;Newton’s method,Nonlinear equation
958;Lithuania;324;0;Basic;Linear Algebra;Linear Transformations;Linearity,Linear application
1091;Lithuania;306;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism,Injective linear application
473;Portugal;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1087;Portugal;455;0;Basic;Linear Algebra;Vector Spaces;Basis
661;Portugal;204;1;Basic;Statistics;Statistics;Student distribution,Hypothesis testing
753;Italy;952;1;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
1082;Portugal;422;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
1220;Slovenia;431;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1334;Portugal;450;0;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
582;Lithuania;696;0;Advanced;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Intersection,Cartesian equations of a line
1517;Portugal;424;0;Advanced;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
626;Italy;253;0;Basic;Differentiation;Derivatives;Logarithmic rule,Trigonometric rules
91;Lithuania;515;0;Advanced;Fundamental Mathematics;Elementary Geometry;Geometrical figures,Quadrilaterals,Rhombus
473;Portugal;292;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism
984;Romania;317;0;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
483;Portugal;93;1;Basic;Differentiation;Derivatives;Product rule,Logarithmic rule,Trigonometric rules
155;Spain;169;1;Basic;Differential Equations;Differential Equations;Linear differential equation
986;Portugal;636;0;Basic;Linear Algebra;Matrices and Determinants;Matrix operations,Matrix multiplication
1045;Lithuania;912;0;Basic;Differentiation;Partial Differentiation;Minimum,Stationary point,Maximum,Saddle point,Optimal solution
890;Portugal;432;1;Advanced;Linear Algebra;Vector Spaces;Linear independence,Subspace,Linear combination,Linear dependence,Span
1219;Portugal;785;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
155;Spain;869;0;Basic;Differential Equations;Differential Equations;Separable variables equation
884;Italy;444;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel
913;Ireland;1212;1;Basic;Statistics;Statistics;Mean,Median,Outliers,Interquartile range
509;Portugal;241;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
625;Russian Federation;509;0;Advanced;Fundamental Mathematics;Elementary Geometry;Symmetry
359;Portugal;443;1;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism
750;Portugal;415;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
510;Portugal;795;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Square roots,Simplify expressions
311;Ireland;647;0;Basic;Linear Algebra;Matrices and Determinants;Matrix multiplication,Symmetric matrix,Matrix operations
1253;Portugal;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
175;Italy;114;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
974;Portugal;1099;0;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Newton’s interpolation
509;Portugal;246;1;Basic;Linear Algebra;Linear Systems;Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system
1487;Portugal;1170;1;Basic;Integration;Definite Integrals;Fundamental theorem of Calculus,Direct integrals 
407;Portugal;337;1;Basic;Complex Numbers;Complex Numbers;Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number
840;Italy;435;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
359;Portugal;447;1;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
91;Lithuania;521;1;Basic;Fundamental Mathematics;Elementary Geometry;Polygons,Regular polygon
57;Portugal;188;0;Basic;Statistics;Statistics;Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median
64;Portugal;667;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
174;Portugal;329;0;Basic;Complex Numbers;Complex Numbers;Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number
524;Portugal;104;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
451;Portugal;294;1;Basic;Probability ;Probability ;Event,Axioms of probability,Sample space
457;Portugal;359;1;Advanced;Complex Numbers;Complex Numbers;Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form
57;Portugal;177;0;Basic;Statistics;Statistics;Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile
80;Lithuania;371;1;Basic;Fundamental Mathematics;Elementary Geometry;Quadrilaterals,Geometrical figures
451;Portugal;302;0;Basic;Probability ;Probability ;Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability
359;Portugal;358;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part
1333;Portugal;417;0;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
168;Portugal;376;0;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation
52;Portugal;129;0;Basic;Linear Algebra;Matrices and Determinants;Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank
890;Portugal;378;0;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
204;Lithuania;102;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
215;Lithuania;834;1;Basic;Differential Equations;Differential Equations;Linear differential equation
376;Portugal;346;0;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
1292;Italy;602;0;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial
91;Lithuania;384;0;Basic;Linear Algebra;Linear Transformations;Linearity,Invertible linear operator,Composition of linear applications,Linear application
1056;Lithuania;885;1;Basic;Differentiation;Partial Differentiation;Second order,First order
1420;Portugal;293;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
422;Portugal;641;1;Basic;Linear Algebra;Matrices and Determinants;Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations
626;Italy;923;1;Basic;Real Functions of a single variable;Domain, Image and Graphics;Domain
57;Portugal;184;1;Basic;Statistics;Statistics;Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency
1311;Italy;382;0;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
80;Lithuania;798;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
71;Portugal;250;1;Basic;Differentiation;Derivatives;Quotient rule,Product rule,Trigonometric rules
986;Portugal;420;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
462;Lithuania;636;0;Basic;Linear Algebra;Matrices and Determinants;Matrix operations,Matrix multiplication
851;Italy;1012;0;Basic;Optimization;Nonlinear Optimization;Optimal solution,Stationary point,Minimum,Maximum,Saddle point
1220;Slovenia;319;1;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation,Linear application,Linearity
239;Lithuania;266;0;Basic;Differentiation;Partial Differentiation;First order,Second order
898;Portugal;454;1;Basic;Linear Algebra;Vector Spaces;Basis
700;Portugal;179;1;Basic;Statistics;Statistics;Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile
876;Italy;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
855;Italy;223;0;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
28;Portugal;552;0;Basic;Analytic Geometry;Analytic Geometry;Orthogonality,Cross product
91;Lithuania;421;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1080;Portugal;785;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1273;Portugal;446;0;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
636;Italy;190;0;Basic;Statistics;Statistics;Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median
1557;Portugal;673;0;Advanced;Real Functions of a single variable;Limits and Continuity;Limits,Indeterminate forms
219;Lithuania;789;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
397;Portugal;357;0;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers
69;Portugal;571;1;Basic;Analytic Geometry;Analytic Geometry;Intersection,Cartesian equations of a plane
1324;Italy;415;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1182;Portugal;435;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
311;Ireland;127;1;Basic;Linear Algebra;Matrices and Determinants;Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank
1147;Portugal;313;0;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
776;Portugal;570;0;Advanced;Analytic Geometry;Analytic Geometry;Distance,Orthogonal projection,Orthogonality
1542;Portugal;1117;1;Basic;Numerical Methods;Numerical Methods;Newton’s interpolation,Polynomial interpolation
1182;Portugal;431;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1350;Portugal;459;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
1322;Italy;289;1;Advanced;Linear Algebra;Linear Transformations;Kernel,Isomorphism,Injective linear application,Range
376;Portugal;1060;1;Basic;Complex Numbers;Complex Numbers;Complex plane 
1039;Lithuania;268;0;Basic;Differentiation;Partial Differentiation;First order,Second order
1525;Portugal;774;0;Advanced;Probability ;Probability ;Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability
955;Lithuania;378;1;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
1252;Portugal;422;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
85;Lithuania;1141;1;Basic;Set Theory;Set Theory;Subset
524;Portugal;367;1;Basic;Fundamental Mathematics;Elementary Geometry;Geometrical figures
1220;Slovenia;455;1;Basic;Linear Algebra;Vector Spaces;Basis
884;Italy;321;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1242;Portugal;454;0;Basic;Linear Algebra;Vector Spaces;Basis
1427;Portugal;385;1;Advanced;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
955;Lithuania;382;0;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
1322;Italy;426;1;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
1194;Portugal;780;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Range,Kernel,Isomorphism
913;Ireland;439;0;Advanced;Linear Algebra;Vector Spaces;Span,Subspace,Linear combination
483;Portugal;356;0;Advanced;Complex Numbers;Complex Numbers;Operations with complex numbers
1277;Portugal;288;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
986;Portugal;779;1;Advanced;Linear Algebra;Linear Transformations;Rotation,Linear application
625;Russian Federation;787;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions,Logarithmic function
1561;Portugal;411;1;Advanced;Integration;Integration Techniques;Integration by parts
1076;Portugal;457;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
913;Ireland;961;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system
913;Ireland;773;1;Basic;Probability ;Probability ;Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability
1219;Portugal;785;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
986;Portugal;306;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism,Injective linear application
955;Lithuania;1213;0;Basic;Numerical Methods;Numerical Methods;Mean,Median,Outliers,Interquartile range
451;Portugal;299;0;Basic;Probability ;Probability ;Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability
884;Italy;383;1;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
1023;Lithuania;457;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
468;Portugal;785;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
37;Portugal;1123;1;Basic;Integration;Integration Techniques;Integration by parts,Substitution,Simple integration,Direct inspection
876;Italy;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1220;Slovenia;431;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1093;Portugal;383;1;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
509;Portugal;776;1;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
1339;Portugal;306;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism,Injective linear application
168;Portugal;449;1;Advanced;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis
1322;Italy;438;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
457;Portugal;354;1;Advanced;Complex Numbers;Complex Numbers;Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas
840;Italy;377;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel,Matrix of a linear transformation
52;Portugal;118;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1479;Portugal;1004;0;Basic;Optimization;Nonlinear Optimization;Minimum,Stationary point,Maximum,Saddle point,Optimal solution
1077;Portugal;449;0;Advanced;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis
349;Portugal;272;0;Basic;Differentiation;Derivatives;Trigonometric rules,Product rule
1292;Italy;630;0;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue
525;Portugal;461;0;Basic;Fundamental Mathematics;Elementary Geometry;Platonic solids
1220;Slovenia;433;0;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
1023;Lithuania;455;1;Basic;Linear Algebra;Vector Spaces;Basis
168;Portugal;443;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism
91;Lithuania;510;1;Advanced;Fundamental Mathematics;Elementary Geometry;Symmetry
1427;Portugal;278;1;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1037;Lithuania;268;0;Basic;Differentiation;Partial Differentiation;First order,Second order
1047;Lithuania;258;0;Basic;Differentiation;Partial Differentiation;First order,Gradient
414;Portugal;360;1;Basic;Complex Numbers;Complex Numbers;Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form
1565;Portugal;981;0;Basic;Real Functions of a single variable;Limits and Continuity;Limits
28;Portugal;229;1;Basic;Linear Algebra;Linear Systems;Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system
634;Italy;784;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1220;Slovenia;454;0;Basic;Linear Algebra;Vector Spaces;Basis
1543;Portugal;1099;1;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Newton’s interpolation
175;Italy;338;0;Basic;Complex Numbers;Complex Numbers;Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form
1182;Portugal;437;0;Basic;Linear Algebra;Linear Transformations;Linear application,Composition of linear applications
201;Lithuania;114;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1076;Portugal;422;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
1542;Portugal;1094;1;Basic;Numerical Methods;Numerical Methods;Newton’s method,Nonlinear equation
1252;Portugal;452;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
91;Lithuania;515;1;Advanced;Fundamental Mathematics;Elementary Geometry;Geometrical figures,Quadrilaterals,Rhombus
1241;Portugal;423;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1023;Lithuania;455;1;Basic;Linear Algebra;Vector Spaces;Basis
91;Lithuania;419;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Dimension
511;Portugal;793;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Square roots,Simplify expressions
1288;Portugal;457;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
455;Portugal;545;1;Basic;Integration;Double Integration;y-simple region (type I),Area ,Cartesian coordinates
168;Portugal;324;0;Basic;Linear Algebra;Linear Transformations;Linearity,Linear application
61;Portugal;160;1;Basic;Differential Equations;Differential Equations;Linear differential equation
1044;Lithuania;999;1;Basic;Differentiation;Partial Differentiation;First order
511;Portugal;239;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
168;Portugal;296;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
840;Italy;287;0;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1288;Portugal;416;1;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
1220;Slovenia;454;1;Basic;Linear Algebra;Vector Spaces;Basis
28;Portugal;703;0;Basic;Analytic Geometry;Analytic Geometry;Orthogonality,Scalar product
128;Lithuania;341;0;Basic;Complex Numbers;Complex Numbers;Operations with complex numbers
1271;Portugal;292;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism
174;Portugal;721;0;Basic;Differentiation;Derivatives;Quotient rule,Trigonometric rules
204;Lithuania;798;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1284;Italy;325;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
769;Portugal;353;0;Basic;Complex Numbers;Complex Numbers;Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas
414;Portugal;92;0;Basic;Linear Algebra;Linear Systems;Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system
128;Lithuania;353;0;Basic;Complex Numbers;Complex Numbers;Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas
167;Portugal;419;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Dimension
1173;Portugal;379;0;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
376;Portugal;438;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1220;Slovenia;435;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
669;Portugal;185;1;Basic;Statistics;Statistics;Frequency,Quantile,Mean,Variance,Median,Mode
1229;Portugal;422;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
128;Lithuania;644;0;Basic;Linear Algebra;Matrices and Determinants;Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations
986;Portugal;440;1;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Subspace,Linear combination,Dimension
967;Portugal;452;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
855;Italy;951;1;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
1505;Portugal;1009;0;Basic;Optimization;Linear Optimization;Transportation problem,Excel solver add-in,Simplex method,Linear programming
986;Portugal;444;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1158;Portugal;438;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
988;Portugal;227;1;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
131;Lithuania;469;1;Advanced;Graph Theory;Graph Theory;Adjacency matrix,Simple graph
71;Portugal;334;1;Basic;Complex Numbers;Complex Numbers;Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form
1220;Slovenia;457;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
1288;Portugal;435;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
415;Portugal;360;0;Basic;Complex Numbers;Complex Numbers;Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form
1220;Slovenia;425;0;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
979;Portugal;1105;0;Basic;Numerical Methods;Numerical Methods;Newton’s method,Nonlinear equation
639;Italy;179;0;Basic;Statistics;Statistics;Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile
1565;Portugal;685;0;Basic;Real Functions of a single variable;Limits and Continuity;Indeterminate forms,Limits
967;Portugal;446;1;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
851;Italy;429;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
852;Italy;227;0;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
1080;Portugal;375;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1219;Portugal;437;0;Basic;Linear Algebra;Linear Transformations;Linear application,Composition of linear applications
1220;Slovenia;443;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism
985;Romania;796;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Logarithmic function,Exponential function,Simplify expressions
972;Portugal;776;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
591;Portugal;326;0;Basic;Complex Numbers;Complex Numbers;Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part
375;Portugal;346;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
1247;Portugal;647;1;Basic;Linear Algebra;Matrices and Determinants;Matrix multiplication,Symmetric matrix,Matrix operations
406;Portugal;243;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
1195;Portugal;437;1;Basic;Linear Algebra;Linear Transformations;Linear application,Composition of linear applications
1220;Slovenia;428;1;Basic;Linear Algebra;Vector Spaces;Change-of-basis matrix,Span,Basis,Dimension
731;Portugal;177;1;Basic;Statistics;Statistics;Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile
411;Portugal;336;0;Basic;Complex Numbers;Complex Numbers;Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part
176;Italy;253;0;Basic;Differentiation;Derivatives;Logarithmic rule,Trigonometric rules
576;Ireland;571;0;Basic;Analytic Geometry;Analytic Geometry;Intersection,Cartesian equations of a plane
1023;Lithuania;431;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
128;Lithuania;637;1;Basic;Linear Algebra;Matrices and Determinants;Determinant computation,Determinant properties
1219;Portugal;785;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1538;Portugal;1088;0;Basic;Numerical Methods;Numerical Methods;First order,Second order
81;Lithuania;553;0;Basic;Analytic Geometry;Analytic Geometry;Orthogonality,Distance,Scalar product
986;Portugal;783;1;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Isomorphism
1220;Slovenia;313;1;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
451;Portugal;322;0;Advanced;Probability ;Probability ;Conditional probability,Bayes' Theorem
81;Lithuania;372;1;Advanced;Graph Theory;Graph Theory;Adjacency matrix,Simple graph,Path,Shortest path
913;Ireland;439;1;Advanced;Linear Algebra;Vector Spaces;Span,Subspace,Linear combination
1220;Slovenia;425;1;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
953;Lithuania;185;1;Basic;Statistics;Statistics;Frequency,Quantile,Mean,Variance,Median,Mode
175;Italy;335;1;Basic;Complex Numbers;Complex Numbers;Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form
352;Portugal;142;0;Basic;Differentiation;Derivatives;Trigonometric rules,Product rule
1487;Portugal;337;0;Basic;Complex Numbers;Complex Numbers;Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number
1220;Slovenia;435;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
37;Portugal;1202;1;Basic;Integration;Definite Integrals;Fundamental theorem of Calculus,Volume of revolution
1087;Portugal;457;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
311;Ireland;127;1;Basic;Linear Algebra;Matrices and Determinants;Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank
969;Romania;430;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
91;Lithuania;439;1;Advanced;Linear Algebra;Vector Spaces;Span,Subspace,Linear combination
90;Lithuania;339;0;Basic;Complex Numbers;Complex Numbers;Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation
1146;Portugal;455;0;Basic;Linear Algebra;Vector Spaces;Basis
1220;Slovenia;377;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel,Matrix of a linear transformation
1229;Portugal;375;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1322;Italy;321;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
415;Portugal;650;1;Basic;Linear Algebra;Matrices and Determinants;Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties
483;Portugal;358;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part
91;Lithuania;354;1;Advanced;Complex Numbers;Complex Numbers;Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas
483;Portugal;262;1;Basic;Differentiation;Partial Differentiation;First order
1220;Slovenia;431;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1220;Slovenia;421;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
36;Portugal;421;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1375;Portugal;382;0;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
966;Portugal;780;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Range,Kernel,Isomorphism
972;Portugal;382;0;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
91;Lithuania;446;0;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
972;Portugal;313;0;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
552;Portugal;983;0;Advanced;Real Functions of a single variable;Limits and Continuity;Limits,Indeterminate forms
1081;Portugal;277;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1220;Slovenia;435;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
854;Italy;952;1;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
57;Portugal;179;0;Basic;Statistics;Statistics;Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile
986;Portugal;447;0;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
1292;Italy;778;0;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
407;Portugal;341;1;Basic;Complex Numbers;Complex Numbers;Operations with complex numbers
91;Lithuania;519;1;Basic;Fundamental Mathematics;Elementary Geometry;Polygons
1517;Portugal;620;1;Advanced;Linear Algebra;Eigenvalues and Eigenvectors;Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum
91;Lithuania;464;1;Basic;Fundamental Mathematics;Elementary Geometry;Polygons,Geometrical figures
663;Portugal;191;0;Basic;Statistics;Statistics;Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance
885;Italy;383;0;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
455;Portugal;138;0;Basic;Linear Algebra;Matrices and Determinants;Matrix operations,Determinant computation,Determinant properties,Inverse matrix 
189;Portugal;735;0;Basic;Differentiation;Derivatives;Product rule,Trigonometric rules,Logarithmic rule
175;Italy;1108;1;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Lagrange's interpolation
955;Lithuania;417;0;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
1148;Portugal;415;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1340;Portugal;1098;1;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Newton’s interpolation
411;Portugal;330;0;Basic;Complex Numbers;Complex Numbers;Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number
1504;Portugal;360;0;Basic;Complex Numbers;Complex Numbers;Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form
411;Portugal;621;0;Advanced;Linear Algebra;Eigenvalues and Eigenvectors;Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum
979;Portugal;949;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
199;Lithuania;665;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
885;Italy;454;0;Basic;Linear Algebra;Vector Spaces;Basis
625;Russian Federation;796;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Logarithmic function,Exponential function,Simplify expressions
918;Portugal;257;0;Basic;Differentiation;Partial Differentiation;Gradient,First order
1197;Slovenia;427;1;Advanced;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
1023;Lithuania;452;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
720;Portugal;211;1;Basic;Statistics;Statistics;Linear regression
61;Portugal;156;0;Basic;Differential Equations;Differential Equations;Linear differential equation
128;Lithuania;635;0;Basic;Linear Algebra;Matrices and Determinants;Determinant computation,Determinant properties,Inverse matrix ,Matrix operations
165;Portugal;661;0;Basic;Differentiation;Derivatives;Logarithmic rule,Product rule,Trigonometric rules
851;Italy;1013;1;Basic;Optimization;Nonlinear Optimization;Optimal solution,Stationary point,Minimum,Maximum,Saddle point
352;Portugal;252;0;Basic;Differentiation;Derivatives;Quotient rule,Product rule,Power rule,Trigonometric rules
1220;Slovenia;444;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel
642;Italy;204;1;Basic;Statistics;Statistics;Student distribution,Hypothesis testing
483;Portugal;424;1;Advanced;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1147;Portugal;431;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1044;Lithuania;886;0;Advanced;Differentiation;Partial Differentiation;Second order,First order
351;Portugal;685;1;Basic;Real Functions of a single variable;Limits and Continuity;Indeterminate forms,Limits
598;Italy;236;0;Advanced;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
576;Ireland;576;0;Advanced;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Cartesian equations of a line
840;Italy;324;1;Basic;Linear Algebra;Linear Transformations;Linearity,Linear application
1087;Portugal;434;1;Basic;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
1220;Slovenia;435;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
374;Portugal;616;0;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvalue,Eigenvector
509;Portugal;666;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
167;Portugal;780;1;Basic;Linear Algebra;Linear Transformations;Injective linear application,Range,Kernel,Isomorphism
470;Portugal;348;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Complex plane ,Real part
174;Portugal;254;1;Basic;Differentiation;Derivatives;Logarithmic rule,Power rule,Trigonometric rules
57;Portugal;1123;1;Basic;Integration;Integration Techniques;Integration by parts,Substitution,Simple integration,Direct inspection
376;Portugal;345;0;Advanced;Complex Numbers;Complex Numbers;Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas
884;Italy;375;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
988;Portugal;420;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
1322;Italy;323;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
979;Portugal;1092;0;Basic;Numerical Methods;Numerical Methods;Newton’s method,Nonlinear equation
972;Portugal;415;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
168;Portugal;324;0;Basic;Linear Algebra;Linear Transformations;Linearity,Linear application
394;Portugal;444;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1324;Italy;443;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism
876;Italy;375;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1109;Portugal;296;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1037;Lithuania;256;0;Basic;Differentiation;Partial Differentiation;First order
90;Lithuania;338;0;Basic;Complex Numbers;Complex Numbers;Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form
1428;Portugal;437;1;Basic;Linear Algebra;Linear Transformations;Linear application,Composition of linear applications
1564;Portugal;923;0;Basic;Real Functions of a single variable;Domain, Image and Graphics;Domain
91;Lithuania;439;1;Advanced;Linear Algebra;Vector Spaces;Span,Subspace,Linear combination
509;Portugal;433;1;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
1247;Portugal;948;0;Basic;Linear Algebra;Linear Systems;Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system
1049;Lithuania;261;0;Basic;Differentiation;Partial Differentiation;First order,Gradient
1146;Portugal;459;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
41;Portugal;569;0;Basic;Analytic Geometry;Analytic Geometry;Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity
1557;Portugal;923;0;Basic;Real Functions of a single variable;Domain, Image and Graphics;Domain
1220;Slovenia;291;1;Basic;Linear Algebra;Linear Transformations;Kernel,Isomorphism,Injective linear application,Range
81;Lithuania;571;0;Basic;Analytic Geometry;Analytic Geometry;Intersection,Cartesian equations of a plane
1220;Slovenia;277;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
91;Lithuania;450;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
600;Italy;652;1;Basic;Linear Algebra;Matrices and Determinants;Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties
37;Portugal;341;0;Basic;Complex Numbers;Complex Numbers;Operations with complex numbers
1300;Italy;435;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
524;Portugal;118;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
91;Lithuania;463;1;Basic;Fundamental Mathematics;Elementary Geometry;Polyhedrons,Geometric solids,Classification of geometric solids
1220;Slovenia;296;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
898;Portugal;428;0;Basic;Linear Algebra;Vector Spaces;Change-of-basis matrix,Span,Basis,Dimension
625;Russian Federation;515;0;Advanced;Fundamental Mathematics;Elementary Geometry;Geometrical figures,Quadrilaterals,Rhombus
483;Portugal;461;1;Basic;Fundamental Mathematics;Elementary Geometry;Platonic solids
1481;Portugal;558;1;Basic;Analytic Geometry;Analytic Geometry;Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line
955;Lithuania;433;0;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
753;Italy;259;0;Basic;Differentiation;Partial Differentiation;First order,Gradient
509;Portugal;948;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system
351;Portugal;114;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
576;Ireland;570;1;Advanced;Analytic Geometry;Analytic Geometry;Distance,Orthogonal projection,Orthogonality
704;Portugal;179;1;Basic;Statistics;Statistics;Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile
91;Lithuania;458;1;Basic;Fundamental Mathematics;Elementary Geometry;Polygons,Geometric transformations,Geometrical figures
854;Italy;245;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
969;Romania;446;1;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
168;Portugal;782;0;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
689;Portugal;818;0;Basic;Optimization;Nonlinear Optimization;Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution
1229;Portugal;454;1;Basic;Linear Algebra;Vector Spaces;Basis
967;Portugal;317;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
376;Portugal;347;0;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
1481;Portugal;560;1;Basic;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Orthogonality
1289;Italy;780;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Range,Kernel,Isomorphism
853;Italy;90;1;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
1135;Portugal;306;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism,Injective linear application
899;Lithuania;436;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
509;Portugal;428;1;Basic;Linear Algebra;Vector Spaces;Change-of-basis matrix,Span,Basis,Dimension
1289;Italy;419;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Dimension
840;Italy;319;1;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation,Linear application,Linearity
1219;Portugal;379;0;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
91;Lithuania;431;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1220;Slovenia;377;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel,Matrix of a linear transformation
28;Portugal;90;1;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
642;Italy;202;0;Basic;Statistics;Statistics;Student distribution,Confidence interval,Point estimate,Hypothesis testing
1220;Slovenia;457;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
1308;Italy;382;1;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
636;Italy;183;0;Basic;Statistics;Statistics;Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode
956;Portugal;433;1;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
884;Italy;776;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
1350;Portugal;426;1;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
1132;Portugal;293;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
37;Portugal;1031;1;Advanced;Differentiation;Partial Differentiation;First order
483;Portugal;441;0;Advanced;Linear Algebra;Vector Spaces;Basis,Span
1219;Portugal;296;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1411;Portugal;380;1;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
72;Portugal;423;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
91;Lithuania;505;1;Basic;Fundamental Mathematics;Elementary Geometry;Triangles
165;Portugal;878;0;Basic;Differentiation;Partial Differentiation;Second order,First order
966;Portugal;776;0;Basic;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
69;Portugal;549;1;Basic;Analytic Geometry;Analytic Geometry;Distance,Cross product,Orthogonality
599;Italy;605;0;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial
1220;Slovenia;423;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
37;Portugal;268;1;Basic;Differentiation;Partial Differentiation;First order,Second order
723;Portugal;188;0;Basic;Statistics;Statistics;Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median
1340;Portugal;1103;1;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Newton’s interpolation
422;Portugal;326;0;Basic;Complex Numbers;Complex Numbers;Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part
684;Ireland;193;0;Basic;Statistics;Statistics;Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean
80;Lithuania;275;0;Basic;Differentiation;Derivatives;Logarithmic rule,Product rule,Trigonometric rules
411;Portugal;852;1;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Continuity,Limits
1220;Slovenia;433;1;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
1241;Portugal;452;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
128;Lithuania;328;0;Basic;Complex Numbers;Complex Numbers;Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number
1284;Italy;439;1;Advanced;Linear Algebra;Vector Spaces;Span,Subspace,Linear combination
406;Portugal;795;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Square roots,Simplify expressions
510;Portugal;521;1;Basic;Fundamental Mathematics;Elementary Geometry;Polygons,Regular polygon
1163;Portugal;453;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1277;Portugal;380;1;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1505;Portugal;285;0;Basic;Optimization;Linear Optimization;Linear programming
411;Portugal;137;0;Basic;Linear Algebra;Matrices and Determinants;Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation
634;Italy;321;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
411;Portugal;241;0;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
1087;Portugal;459;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
1081;Portugal;323;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
419;Portugal;353;0;Basic;Complex Numbers;Complex Numbers;Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas
988;Portugal;89;1;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
1179;Portugal;444;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
972;Portugal;439;0;Advanced;Linear Algebra;Vector Spaces;Span,Subspace,Linear combination
1319;Italy;433;1;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
1273;Portugal;423;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
375;Portugal;352;1;Advanced;Complex Numbers;Complex Numbers;Conjugate number,Operations with complex numbers
91;Lithuania;853;1;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Continuity,Limits
1286;Italy;422;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
415;Portugal;128;0;Advanced;Linear Algebra;Matrices and Determinants;Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon
751;Lithuania;558;0;Basic;Analytic Geometry;Analytic Geometry;Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line
972;Portugal;384;0;Basic;Linear Algebra;Linear Transformations;Linearity,Invertible linear operator,Composition of linear applications,Linear application
979;Portugal;1088;1;Basic;Numerical Methods;Numerical Methods;First order,Second order
468;Portugal;296;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1197;Slovenia;451;1;Advanced;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
168;Portugal;444;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1277;Portugal;292;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism
1220;Slovenia;383;1;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
1525;Portugal;766;0;Basic;Probability ;Probability ;Event,Sample space
91;Lithuania;326;1;Basic;Complex Numbers;Complex Numbers;Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part
1220;Slovenia;379;0;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
1220;Slovenia;784;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1543;Portugal;1098;0;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Newton’s interpolation
899;Lithuania;444;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1196;Slovenia;433;0;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
1561;Portugal;833;0;Advanced;Differential Equations;Differential Equations;Linear differential equation
41;Portugal;191;1;Basic;Statistics;Statistics;Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance
81;Lithuania;547;1;Basic;Analytic Geometry;Analytic Geometry;Cross product,Orthogonality,Scalar product
913;Ireland;446;1;Advanced;Linear Algebra;Vector Spaces;Basis,Span,Change-of-basis matrix
380;Portugal;865;0;Basic;Optimization;Nonlinear Optimization;Lagrange multipliers,Minimum,Maximum,Stationary point
451;Portugal;312;0;Advanced;Probability ;Probability ;Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space
965;Portugal;436;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
855;Italy;219;0;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
890;Portugal;317;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
1220;Slovenia;452;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1517;Portugal;288;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
166;Portugal;563;0;Basic;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Cartesian equations of a line
1044;Lithuania;876;1;Basic;Differentiation;Partial Differentiation;Directional derivative,Gradient,First order,Second order
91;Lithuania;277;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1075;Portugal;885;0;Basic;Differentiation;Partial Differentiation;Second order,First order
855;Italy;233;0;Basic;Linear Algebra;Linear Systems;Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system
626;Italy;717;1;Basic;Differentiation;Derivatives;Trigonometric rules,Quotient rule
1080;Portugal;325;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1070;Portugal;452;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
986;Portugal;419;1;Advanced;Linear Algebra;Vector Spaces;Subspace,Dimension
1220;Slovenia;785;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
470;Portugal;555;0;Basic;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Orthogonality,Locus,Distance
1220;Slovenia;433;1;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
1220;Slovenia;287;1;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation
91;Lithuania;511;1;Basic;Fundamental Mathematics;Elementary Geometry;Triangle inequality,Triangles
969;Romania;459;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
562;Ireland;638;0;Advanced;Linear Algebra;Matrices and Determinants;Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices
840;Italy;438;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
525;Portugal;504;1;Basic;Fundamental Mathematics;Elementary Geometry;Symmetry,Geometric transformations
1220;Slovenia;384;1;Basic;Linear Algebra;Linear Transformations;Linearity,Invertible linear operator,Composition of linear applications,Linear application
422;Portugal;633;0;Advanced;Linear Algebra;Matrices and Determinants;Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation
81;Lithuania;560;0;Basic;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Orthogonality
986;Portugal;380;1;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
988;Portugal;951;1;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
986;Portugal;289;0;Advanced;Linear Algebra;Linear Transformations;Kernel,Isomorphism,Injective linear application,Range
69;Portugal;560;1;Basic;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Orthogonality
972;Portugal;784;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
36;Portugal;1127;1;Basic;Set Theory;Set Theory;Union of sets,Venn diagram,Complement of a set
1158;Portugal;429;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1182;Portugal;454;0;Basic;Linear Algebra;Vector Spaces;Basis
956;Portugal;436;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
986;Portugal;424;1;Advanced;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1220;Slovenia;313;1;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
988;Portugal;230;1;Basic;Linear Algebra;Linear Systems;Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system
1194;Portugal;324;1;Basic;Linear Algebra;Linear Transformations;Linearity,Linear application
1087;Portugal;435;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
1044;Lithuania;267;0;Basic;Differentiation;Partial Differentiation;First order,Second order
349;Portugal;250;1;Basic;Differentiation;Derivatives;Quotient rule,Product rule,Trigonometric rules
509;Portugal;608;1;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvalue,Eigenvector
986;Portugal;778;0;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1179;Portugal;292;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism
69;Portugal;554;0;Basic;Analytic Geometry;Analytic Geometry;Scalar product,Orthogonality,Orthogonal projection,Distance
1220;Slovenia;459;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
64;Portugal;735;1;Basic;Differentiation;Derivatives;Product rule,Trigonometric rules,Logarithmic rule
852;Italy;220;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
406;Portugal;138;0;Basic;Linear Algebra;Matrices and Determinants;Matrix operations,Determinant computation,Determinant properties,Inverse matrix 
406;Portugal;646;0;Basic;Linear Algebra;Matrices and Determinants;Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix
509;Portugal;949;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
651;Ireland;181;0;Basic;Statistics;Statistics;Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode
876;Italy;383;0;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
1132;Portugal;783;0;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Isomorphism
128;Lithuania;635;0;Basic;Linear Algebra;Matrices and Determinants;Determinant computation,Determinant properties,Inverse matrix ,Matrix operations
72;Portugal;292;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism
219;Lithuania;787;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions,Logarithmic function
168;Portugal;450;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
142;Portugal;614;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
351;Portugal;727;0;Advanced;Differentiation;Derivatives;Trigonometric rules,Product rule
510;Portugal;225;0;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
1253;Portugal;296;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
91;Lithuania;524;0;Basic;Fundamental Mathematics;Elementary Geometry;Quadrilaterals,Rhombus
1291;Italy;313;1;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
741;Ireland;193;0;Basic;Statistics;Statistics;Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean
1087;Portugal;422;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
582;Lithuania;345;1;Advanced;Complex Numbers;Complex Numbers;Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas
64;Portugal;108;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1220;Slovenia;435;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
1288;Portugal;415;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
969;Romania;448;1;Advanced;Linear Algebra;Vector Spaces;Subspace
204;Lithuania;104;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1220;Slovenia;317;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
406;Portugal;127;1;Basic;Linear Algebra;Matrices and Determinants;Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank
1517;Portugal;293;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
91;Lithuania;346;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
625;Russian Federation;800;0;Advanced;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Exponential function,Simplify expressions
509;Portugal;317;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
966;Portugal;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1241;Portugal;425;1;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
659;Portugal;80;0;Basic;Statistics;Statistics;Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean
851;Italy;1005;0;Basic;Optimization;Nonlinear Optimization;Minimum,Stationary point,Maximum,Saddle point,Optimal solution
380;Portugal;859;0;Basic;Optimization;Nonlinear Optimization;Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point
64;Portugal;397;1;Basic;Integration;Integration Techniques;Substitution
1402;Portugal;380;1;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
966;Portugal;378;0;Basic;Linear Algebra;Linear Transformations;Invertible linear operator,Isomorphism,Injective linear application
28;Portugal;951;1;Basic;Linear Algebra;Linear Systems;Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system
851;Italy;377;1;Basic;Linear Algebra;Linear Transformations;Range,Kernel,Matrix of a linear transformation
1220;Slovenia;313;1;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
1087;Portugal;416;1;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
913;Ireland;1095;0;Basic;Numerical Methods;Numerical Methods;Newton’s method,Nonlinear equation
979;Portugal;1107;1;Basic;Numerical Methods;Numerical Methods;Newton’s method,Nonlinear equation
405;Portugal;118;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1557;Portugal;927;0;Advanced;Real Functions of a single variable;Domain, Image and Graphics;Planar graph
1505;Portugal;283;0;Basic;Optimization;Linear Optimization;Excel solver add-in,Simplex method,Linear programming
411;Portugal;640;0;Basic;Linear Algebra;Matrices and Determinants;Determinant computation,Determinant properties
37;Portugal;311;1;Basic;Integration;Integration Techniques;Direct inspection,Simple integration
525;Portugal;364;1;Basic;Fundamental Mathematics;Elementary Geometry;Geometric solids,Geometrical figures,Classification of geometric solids
72;Portugal;424;0;Advanced;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1023;Lithuania;459;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
511;Portugal;220;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
483;Portugal;448;1;Advanced;Linear Algebra;Vector Spaces;Subspace
986;Portugal;451;1;Advanced;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
986;Portugal;289;1;Advanced;Linear Algebra;Linear Transformations;Kernel,Isomorphism,Injective linear application,Range
1244;Portugal;450;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
91;Lithuania;415;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
511;Portugal;611;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
510;Portugal;607;0;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvalue,Eigenvector
986;Portugal;381;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
535;Portugal;452;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
483;Portugal;142;1;Basic;Differentiation;Derivatives;Trigonometric rules,Product rule
633;Italy;425;1;Basic;Linear Algebra;Vector Spaces;Subspace,Change-of-basis matrix,Linear combination,Span,Basis
548;Portugal;486;0;Basic;Probability ;Probability ;Event,Addition, multiplication, and total probability rules,Sample space
91;Lithuania;778;0;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
986;Portugal;289;1;Advanced;Linear Algebra;Linear Transformations;Kernel,Isomorphism,Injective linear application,Range
1028;Lithuania;998;0;Basic;Differentiation;Partial Differentiation;First order
1284;Italy;325;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
525;Portugal;519;1;Basic;Fundamental Mathematics;Elementary Geometry;Polygons
1087;Portugal;452;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
633;Italy;638;1;Advanced;Linear Algebra;Matrices and Determinants;Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices
1220;Slovenia;455;0;Basic;Linear Algebra;Vector Spaces;Basis
1077;Portugal;319;1;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation,Linear application,Linearity
1517;Portugal;857;1;Advanced;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial
1427;Portugal;288;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
913;Ireland;297;0;Basic;Probability ;Probability ;Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability
1375;Portugal;779;1;Advanced;Linear Algebra;Linear Transformations;Rotation,Linear application
899;Lithuania;431;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
851;Italy;155;0;Basic;Optimization;Nonlinear Optimization;Lagrange multipliers,Maximum,Minimum
663;Portugal;183;1;Basic;Statistics;Statistics;Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode
468;Portugal;321;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
1481;Portugal;298;0;Basic;Probability ;Probability ;Event,Axioms of probability
913;Ireland;487;0;Basic;Probability ;Probability ;Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space
1331;Portugal;431;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
80;Lithuania;666;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
973;Portugal;224;1;Basic;Linear Algebra;Linear Systems;Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system
1319;Italy;383;0;Basic;Linear Algebra;Linear Transformations;Linearity,Matrix of a linear transformation,Linear application
899;Lithuania;436;0;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
600;Italy;125;0;Basic;Linear Algebra;Matrices and Determinants;Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication
751;Lithuania;699;1;Basic;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line
37;Portugal;401;1;Basic;Integration;Integration Techniques;Direct inspection,Simple integration
1288;Portugal;292;0;Advanced;Linear Algebra;Linear Transformations;Range,Kernel,Isomorphism
1220;Slovenia;452;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
376;Portugal;454;0;Basic;Linear Algebra;Vector Spaces;Basis
1322;Italy;421;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1023;Lithuania;452;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1044;Lithuania;882;0;Advanced;Differentiation;Partial Differentiation;Gradient,First order,Directional derivative
168;Portugal;313;0;Basic;Linear Algebra;Vector Spaces;Span,Linear combination,Subspace
1241;Portugal;422;0;Basic;Linear Algebra;Vector Spaces;Linear combination,Span
1518;Portugal;104;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
1080;Portugal;457;1;Basic;Linear Algebra;Vector Spaces;Linear combination,Subspace,Span
1288;Portugal;278;1;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1308;Italy;450;0;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
625;Russian Federation;458;1;Basic;Fundamental Mathematics;Elementary Geometry;Polygons,Geometric transformations,Geometrical figures
91;Lithuania;384;1;Basic;Linear Algebra;Linear Transformations;Linearity,Invertible linear operator,Composition of linear applications,Linear application
884;Italy;321;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
483;Portugal;254;0;Basic;Differentiation;Derivatives;Logarithmic rule,Power rule,Trigonometric rules
41;Portugal;681;1;Basic;Real Functions of a single variable;Limits and Continuity;Indeterminate forms,Limits
1220;Slovenia;296;1;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
72;Portugal;438;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
177;Italy;798;1;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
913;Ireland;761;1;Basic;Probability ;Probability ;Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability
633;Italy;420;1;Advanced;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
72;Portugal;334;0;Basic;Complex Numbers;Complex Numbers;Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form
958;Lithuania;423;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
403;Portugal;782;0;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1291;Italy;450;0;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
1220;Slovenia;317;0;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
1220;Slovenia;376;0;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1247;Portugal;347;0;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
85;Lithuania;1141;1;Basic;Set Theory;Set Theory;Subset
972;Portugal;436;1;Basic;Linear Algebra;Vector Spaces;Euclidean spaces,Orthogonal basis,Basis
509;Portugal;429;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1132;Portugal;278;1;Advanced;Linear Algebra;Linear Transformations;Matrix of a linear transformation
899;Lithuania;317;1;Basic;Linear Algebra;Vector Spaces;Subspace,Span,Linear combination
37;Portugal;1214;1;Basic;Numerical Methods;Numerical Methods;Mean,Median,Outliers,Interquartile range
37;Portugal;142;1;Basic;Differentiation;Derivatives;Trigonometric rules,Product rule
1292;Italy;593;1;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue
1023;Lithuania;433;1;Basic;Linear Algebra;Vector Spaces;Subspace,Dimension
155;Spain;168;1;Basic;Differential Equations;Differential Equations;Linear differential equation
1411;Portugal;288;1;Advanced;Linear Algebra;Linear Transformations;Range,Kernel
884;Italy;319;0;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation,Linear application,Linearity
955;Lithuania;417;0;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
81;Lithuania;365;0;Advanced;Graph Theory;Graph Theory;Adjacency matrix,Simple graph
1517;Portugal;382;0;Advanced;Linear Algebra;Linear Transformations;Injective linear application,Invertible linear operator,Isomorphism
1220;Slovenia;444;0;Basic;Linear Algebra;Linear Transformations;Range,Kernel
1340;Portugal;1113;1;Basic;Numerical Methods;Numerical Methods;Newton’s interpolation,Polynomial interpolation
972;Portugal;427;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
753;Italy;780;1;Basic;Linear Algebra;Linear Transformations;Injective linear application,Range,Kernel,Isomorphism
913;Ireland;188;0;Basic;Statistics;Statistics;Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median
855;Italy;91;0;Basic;Linear Algebra;Linear Systems;Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system
1081;Portugal;424;0;Advanced;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1543;Portugal;1101;0;Basic;Numerical Methods;Numerical Methods;Polynomial interpolation,Newton’s interpolation
351;Portugal;120;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
535;Portugal;429;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
1146;Portugal;429;0;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
979;Portugal;1251;1;Basic;Numerical Methods;Numerical Methods;Newton’s interpolation,Polynomial interpolation
509;Portugal;454;1;Basic;Linear Algebra;Vector Spaces;Basis
1244;Portugal;415;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
884;Italy;291;0;Basic;Linear Algebra;Linear Transformations;Kernel,Isomorphism,Injective linear application,Range
1525;Portugal;775;0;Advanced;Probability ;Probability ;Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability
1242;Portugal;454;1;Basic;Linear Algebra;Vector Spaces;Basis
890;Portugal;376;0;Basic;Linear Algebra;Linear Transformations;Matrix of a linear transformation
1079;Portugal;431;1;Basic;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
208;Lithuania;645;1;Basic;Linear Algebra;Matrices and Determinants;Matrix rank,Linearly independent rows
1220;Slovenia;417;1;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
576;Ireland;550;0;Advanced;Analytic Geometry;Analytic Geometry;Cross product,Orthogonality
1517;Portugal;430;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Linear combination,Span
890;Portugal;441;1;Advanced;Linear Algebra;Vector Spaces;Basis,Span
37;Portugal;596;1;Basic;Linear Algebra;Eigenvalues and Eigenvectors;Eigenvector,Eigenvalue
177;Italy;250;0;Basic;Differentiation;Derivatives;Quotient rule,Product rule,Trigonometric rules
174;Portugal;328;1;Basic;Complex Numbers;Complex Numbers;Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number
72;Portugal;427;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Basis,Dimension
406;Portugal;667;0;Basic;Fundamental Mathematics;Algebraic expressions, Equations, and Inequalities;Simplify expressions
239;Lithuania;93;0;Basic;Differentiation;Derivatives;Product rule,Logarithmic rule,Trigonometric rules
955;Lithuania;419;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Dimension
71;Portugal;651;0;Basic;Linear Algebra;Matrices and Determinants;Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties
91;Lithuania;419;0;Advanced;Linear Algebra;Vector Spaces;Subspace,Dimension
1026;Lithuania;658;0;Basic;Differentiation;Derivatives;Product rule,Trigonometric rules
411;Portugal;393;0;Basic;Integration;Integration Techniques;Direct inspection,Simple integration
972;Portugal;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
840;Italy;780;1;Basic;Linear Algebra;Linear Transformations;Injective linear application,Range,Kernel,Isomorphism
374;Portugal;415;0;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1340;Portugal;1089;1;Basic;Numerical Methods;Numerical Methods;First order,Second order
1044;Lithuania;261;1;Basic;Differentiation;Partial Differentiation;First order,Gradient
535;Portugal;346;1;Advanced;Complex Numbers;Complex Numbers;Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part
37;Portugal;339;1;Basic;Complex Numbers;Complex Numbers;Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation
483;Portugal;352;0;Advanced;Complex Numbers;Complex Numbers;Conjugate number,Operations with complex numbers
91;Lithuania;498;0;Advanced;Fundamental Mathematics;Elementary Geometry;Triangles,Classification of geometrical figures,Geometrical figures
1220;Slovenia;416;1;Basic;Linear Algebra;Vector Spaces;Basis,Dimension
1220;Slovenia;323;0;Basic;Linear Algebra;Linear Transformations;Linear application,Linearity
576;Ireland;561;0;Basic;Analytic Geometry;Analytic Geometry;Cartesian equations of a plane,Cartesian equations of a line
91;Lithuania;453;1;Basic;Linear Algebra;Vector Spaces;Linear independence,Span,Linear dependence
1517;Portugal;448;0;Advanced;Linear Algebra;Vector Spaces;Subspace