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principles of logical reasoning—Affirming the Hypothesis and Chaining ..... Key
themes are discrete mathematical modeling, optimization ..... investigation, look
for answers to this question: How can you ...
while also preparing you for future study of geometry, probability, and data
analysis. ... practice skills to challenging exercises that involve logical reasoning
and problem ... Look for a notetaking strategy at the beginning of each chapter as
.
answer in the box provided, placing one digit in each box and no spaces
between digits. MA.912. .... Look at the triangle ABC. ...... This is not logical
because the original statement doesn't state that only ...
students needing a review of key content in Geometry. Students who ... These
items may be multiple choice or short answer. .... When writing a proof, it is
important to justify each logical step with a reason. .... looking at the picture of the
30°–.
strengthen and build logically upon your current knowledge and skills. ...
advanced tests in algebra and geometry. ... mark on the answer sheet next to any
item you skip over as long as you erase any stray .... get a closer look at the place
kept.
The Geometry teachers in the Tigard-Tualatin School District look forward to
working with you next year. It is our ... conjectures, and building logical
arguments. ... The answer key is meant to be a tool for ...
Key Terms for This Session .... and draw conclusions based more on logic than
intuition. ... What do you think were the key pieces of geometry content in this
activity? .... As you look at the next set of problems, answer these questions: a.
18 questions—Geometry. 40 questions ... algebra, and plane geometry) are
tested, but most of the ... and rules of formal logic are not tested. The test ... An
answer key is provided at the end of ... the characters; we look at the hands as
the man.
Geometry and Measurement. 9 ... It focuses on the key concepts of mathematics
and on the ability to solve ... Many of the problems require the integration of
multiple skills to achieve a solution. ... Logical connectives and quantifiers:
interpret.
Discovering Geometry Logic Lessons. 17 ... Let's take a closer look at some
examples of the logical reasoning used by ...... In Exercises 13–17, use clear
logical reasoning to answer each question. 13.
Today you will be taking the Missouri Geometry Test. This is a test of ... your
answers directly on your answer sheet with a number 2 pencil. .... Look at the
letters below. ... Which of these is the most logical order for the statements and
reasons? |
Holt, rinehart and winston - mat home page, Middle school math, pre-algebra, algebra and geometry lessons. helpful links to middle school math resources on the internet. do a keyword search or select a subject.
Go hrw, We would like to show you a description here but the site won't allow us..
Holt mathematics course 2 | ebay - electronics, cars, fashion, Find great deals on ebay for holt mathematics course 2 in education textbooks. shop with confidence..
Classzone - algebra 2, Welcome to algebra 2. this course will make math come alive with its many intriguing examples of algebra in the world around you, from baseball to theater lighting to.
Holt mcdougal math textbooks - learning things, Holt mcdougal math textbooks for middle school offers comprehensive instruction, assessment, and intervention tools for complete coverage of the common core standards.. |
Summary: These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short problems and exercises that focus on developing a particular skill, often requiring students to draw or interpret sketches and graphs, or reason with math relationships. New to the Second Edition are exercises that provide guided practice for the textbook's Problem-Solving Strategies, focusing in particular on working symbolically.
2nd |
_________________
it took 2minutes to me to choose IMO B; here is my reasoning: A. this is evident from the last line of argument "even though the deepest mathematical......" C. we can find it paraphrasing "The graphical illustrations mathematics teachers use enable students to learn geometry more easily...." d. again same reasoning as in C. e. whole argument support this option. B. i dint find any support in argument for this option.
OA please, please let me know if my reasoning is correct and also please tell the difficulty level of this problem, and is 2mins OK? i found this question little bit difficult.
Last edited by gmatstar10 on 23 Apr 2010, 21:35, edited 1 time in total.Your reasoning is in reversed order. For strengthen questions, you need to doubt the argument and take the stated answer choices as CORRECT to support the conclusion not the other way round.
IMO B.
poohv005 wrote....graphical illustrationsenable students to learngeometry more easily by providing them with an intuitive understanding of geometric concepts, which makes it easier to acquire the ability to manipulate symbols
so having preliminary understanding of math concepts makes it easier to acquire the ability to manipulate symbols .
in B we have controversial reasoning, stating that: thouse who are good in manipulating symbols do not necessarily have any mathematical understanding. - vice versa reasoning. |
a terminal algebra course or to prepare students for college-level algebra, precalculus, or finite math. Elayn Martin-Gay focuses on enhancing a traditional emphasismastering the basicswith innovative pedagogy and a meaningful, integrated learning program. Three goals drive her authorship: increase motivation, build confidence, and encourage mastery of basic skills and concepts. Martin-Gay's unparalleled ability to explain key concepts and build problem-solving skills is enlivened by her ability to relate to students through real-l... MOREife applications that are interesting, relevant and practical. The completely integrated teaching and learning package that accompanies the new editions includes the critically acclaimed MathPro 4.0 Tutorial Software and the popular Lecture Video Series written, directed and performed by Elayn Martin-Gay for maximum student support and learning effectiveness |
Product Description
This program shows students that linear equations help explain changes in the real world. These equations can predict the future, that's right....they can tell us what's happened, what's happening, and what might happen. So get out your crystal ball, and get ready to understand the powers of linear equations.Topics Covered: Slope Horizontal and Vertical Lines Parallel and Perpendicular Lines 3 Forms of Linear Equations Connecting with Graphs Systems of Equations Includes a DVD plus a CD-ROM with teacher's guide, quizzes, graphic organizers and classroom activities. Teaching Systems programs are optimized for classroom use and include "Full Public Performance Rights". Grade |
Author Information
Margaret L. Lial, John Hornsby, Terry McGinnis
Product Details
Edition:
1
ISBN:
9780321507228
Publish Date:
2007-06-02
Publisher:
Addison Wesley
Product Description
Miller/O'Neill's Beginning and Intermediate Algebra is an insightful textbook written by instructors who have first-hand experience with students of developmental mathematics. Through specially designed exercise sets, student-friendly writing, carefully organized page-layout, and helpful hints and tips, Beginning and Intermediate Algebra engages students in their study of mathematics and paves the pathway for success.Written by authorities in Mathematics, Beginning and Intermediate Algebra plus MyMathLab Student Access Kit (4th Edition) by Margaret L. Lial, John Hornsby, and Terry McGinnis provides an excellent foundation for Mathematics studies. Margaret L. Lial, John Hornsby, and Terry McGinnis??™s style is excellently suited towards Mathematics studies, and will teach students the material clearly without overcomplicating the subject. What??™s more, the text is available in the Hardcover format shown above (ISBN 9780321507228), as well as a number of other formats. As of June 2007, this revision raises the bar for Beginning and Intermediate Algebra plus MyMathLab Student Access Kit (4th Edition)??™s high standard of excellence, making sure that it stays one of the foremost Mathematics studies textbooks. |
Product Description
This review book provides two pages of review for each lesson, emphasizing both newly and previously learned skills. The first page reviews concepts taught in the lesson for the day, while the second page reviews concepts taught in previous lessons and provides facts practice. A chapter review, cumulative review, and facts review page are also included at the end of each chapter. Use this book daily or for additional review when needed; pages may be used anytime after the lesson has been taught. 332 pages, softcover. 3rd Edition.
BJU Press is also known as Bob Jones; this is the Bob Jones Math 1 Reviews Book, 3rd Edition.
Product Information
Product Reviews
BJU Math 1 Reviews, Third Edition
4
5
1
1
If you are giving your child a lot of help on the regular workbook pages, then the review book is a good way to give independant work to make sure they understood what you have worked on together.
September 25, 2010 |
Homework
The homeworks are essential in learning linear algebra. They are not a test and you are encouraged to talk to other students about difficult problems-after you have found them difficult. Talking about linear algebra is healthy. But you must write your own solutions.
Exams
There will be three one-hour exams at class times and a final exam. The use of calculators or notes is not permitted during the exams.
Grading
Grading criteria.
ACTIVITIES
PERCENTAGES
Problem sets
15%
Three one-hour exams
45%
Final exam
40%
MATLAB®
Some homework problems will require you to use MATLAB, an important tool for numerical linear algebra. No previous MATLAB experience is required in 18.06. The related resources section has links to information about MATLAB, including a tutorial |
Related Products
Summary
Precalculus: A Problems-Oriented Approach offers a fairly rigorous lead-in to calculus using the right triangle approach to trigonometry. A graphical perspective gives students a visual understanding of concepts. The text may be used with any graphing utility, or with none at all, with equal ease. Modeling provides students with real-world connections to the problems. The author is know for his clear writing style and numerous quality exercises and applications.
Table of Contents
Algebra Background for Precalculus
1
(34)
Sets of Real Numbers
1
(5)
Absolute Value
6
(5)
Polynomials and Factoring
11
(9)
Quadratic Equations
20
(15)
Coordinates, Graphs, and Inequalities
35
(66)
Rectangular Coordinates
35
(12)
Graphs and Equations, A Second Look
47
(10)
Equations of Lines
57
(12)
Symmetry and Graphs
69
(8)
Inequalities
77
(8)
More on Inequalities
85
(16)
Functions
101
(74)
The Definition of a Function
101
(12)
The Graph of a Function
113
(17)
Techniques in Graphing
130
(12)
Methods of Combining Functions. Iteration
142
(12)
Inverse Functions
154
(21)
Polynomial and Rational Functions. Applications to Iteration and Optimization
175
(95)
Linear Functions
175
(14)
Quadratic Functions
189
(9)
More on Iteration. Quadratics and Population Growth
198
(17)
Applied Functions: Setting up Equations
215
(12)
Maximum and Minimum Problems
227
(11)
Polynomial Functions
238
(13)
Rational Functions
251
(19)
Exponential and Logarithmic Functions
270
(87)
Exponential Functions
271
(9)
The Exponential Function y = ex
280
(9)
Logarithmic Functions
289
(12)
Properties of Logarithms
301
(9)
Equations and Inequalities with Logs and Exponents
310
(11)
Compound Interest
321
(11)
Exponential Growth and Decay
332
(25)
Trigonometric Functions of Angles
357
(51)
Trigonometric Functions of Acute Angles
357
(10)
Algebra and the Trigonometric Functions
367
(7)
Right-Triangle Applications
374
(9)
Trigonometric Functions of Angles
383
(12)
Trigonometric Identities
395
(13)
Trigonometric Functions of Real Numbers
408
(75)
Radian Measure
408
(9)
Radian Measure and Geometry
417
(10)
Trigonometric Functions of Real Numbers
427
(11)
Graphs of the Sine and the Cosine Functions
438
(16)
Graphs of y = A sin(Bx - C) and y = A cos(Bx - C)
454
(8)
Simple Harmonic Motion
462
(5)
Graphs of the Tangent and the Reciprocal Functions
467
(16)
Analytical Trigonometry
483
(55)
The Addition Formulas
483
(9)
The Double-Angle Formulas
492
(10)
The Product-to-Sum and Sum-to-Product Formulas
502
(5)
Trigonometric Equations
507
(11)
The Inverse Trigonometric Functions
518
(20)
Additional Topics in Trigonometry
538
(62)
The Law of Sines and the Law of Cosines
538
(15)
Vectors in the Plane, A Geometric Approach
553
(7)
Vectors in the Plane, An Algebraic Approach
560
(8)
Parametric Equations
568
(8)
Introduction to Polar Coordinates
576
(9)
Curves in Polar Coordinates
585
(15)
Systems of Equations
600
(72)
Systems of Two Linear Equations in Two Unknowns
600
(11)
Gaussian Elimination
611
(8)
Matrices
619
(12)
The Inverse of a Square Matrix
631
(7)
Determinants and Cramer's Rule
638
(13)
Nonlinear Systems of Equations
651
(7)
Systems of Inequalities
658
(14)
Analytic Geometry
672
(80)
The Basic Equations
672
(9)
The Parabola
681
(11)
Tangents to Parabolas (Optional)
692
(3)
The Ellipse
695
(15)
The Hyperbola
710
(11)
The Focus-Directrix Property of Conics
721
(9)
The Conics in Polar Coordinates
730
(6)
Rotation of Axes
736
(16)
Roots of Polynomial Equations
752
(70)
The Complex Number System
752
(8)
Division of Polynomials
760
(7)
Roots of Polynomial Equations: The Remainder Theorem and the Factor Theorem |
At Earlham, we strive to teach our students mathematical fundamentals and problem-solving skills that they can apply in a variety of disciplines or in further study of mathematics. Mathematics students may participate in weekly "mathophiles" seminars and informal lunches, attend regional meetings of professional mathematicians, and participate in mathematically related off-campus programs during the academic year or the summer.
The Association for Women in Mathematics (Careers That Count) describes mathematics as "… a powerful tool for solving practical problems and a highly creative field of study, combining logic and precision with intuition and imagination. The basic goal of mathematics is to reveal and explain patterns — whether the pattern appears as electrical impulses in an animal's nervous system, as fluctuations in stock market prices, or as fine detail of an abstract geometric figure."
Earlham College, an independent, residential college, aspires to provide the highest-quality undergraduate education in the liberal arts, including the sciences, shaped by the distinctive perspectives of the Religious Society of Friends (Quakers). |
Precalculus - 4th edition
Summary: PRECALCULUS, Fourth Edition focuses on teaching the essentials of what a student needs to fulfill their PreCalculus requirement and to fully prepare them to succeed in calculus. It provides students with an integrated review of algebra and trigonometry while focusing on essential calculus concepts. Faires and DeFranza prepare students for calculus by providing a solid grounding in analysis and graphing, tools necessary to make a successful transition to calculus. This s...show moretreamlined text provides all the mathematics that students need--it doesn't bog them down in review, or overwhelm them with too much, too soon. The authors are careful to keep this book, unlike many of the PreCalculus books on the market, at a length that can be covered in one term. ...show less
Introduction. The Real Line. Inequalities. Intervals. Absolute Values. The Coordinate Plane. Distance Between Points in the Plane. Circles in the Plane. Completing the Square. Equations and Graphs. Graphs of Equations. Symmetry of a Graph. Using Technology to Graph Equations. Functions. Applications of Functions. Linear Functions. Parallel and Perpendicular Lines. Quadratic Functions. The Standard Form of a Quadratic Equation. The Quadratic FormulaPenntext Downingtown, PA
torn page May have some notes/highlighting, slightly worn covers, general wear/tear. Please contact us if you have any Questions |
This is an accurate facsimile of a handmade 210-page journal. This facsimile can be used by teachers as an example to inspire students to create similar journals.
The journal has 4 types of... More > content:
1. Instructional content which is like lecture notes.
2. Pages where students research various subjects on the Internet. These pages include history, biographies, and topics of interest.
3. Diagrams, charts, and illustrations.
4. Pages of the students' choice where they select the math-related topic.
Although this is an algebra journal, it can be used as an example for any subject where the teacher wants to inspire students to make a similar journal.< Less
This is a complete curriculum for the second semester of algebra. It is designed to be used with the first semester book as well as the student workbook and teacher workbook. All these books are... More > available from Simplified Solutions for Math on LULU. Contact Simplified at ss4math@gmail.com for more information and a complete set of PowerPoint presentations for each lesson, free with purchase. Completely self-contained, ideal for home schooling as well as traditional classrooms"The total revenue function for a product is given by R = 266x, and the total cost function for the same product is C = 2000 + 46x + 2x2, where R and C are each measured in thousands of dollars... More > and x is the number of units produced and sold.
a. Form the profit function for this product from the two given functions.
b. What is the profit when 55 units are produced and sold?
c. How many units must be sold to break even on this product?
"< Less
Chapter 1 only from GOLDen Mathematics: Elementary & Intermediate Algebra. See that book's description for more information. Topics covered: an introduction to algebra and the real number system... More > including working with a calculator. (2 sections, 17 pages)< Less |
....
Show More. Fractals are introduced as self-similar geometric objects and analyzed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia Sets and the Mandelbrot Set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations.The book is richly illustrated and includes over 200 end-of-chapter exercises. A flexible format and a clear and succinct writing style make it a good choice for introductory courses in chaos and fractals |
Vali Nasser
Where to find Vali Nasser online
Where to buy in print
Books
Speed Mathematics Using the Vedic System by Vali Nasser
Price:
$9.90 USD.
Approx. 15,570 words.
Language:
English.
Published on October 30, 2010.
Category:
Nonfiction.
Speed Mathematics using the Vedic system makes learning basic mathematics more rewarding. The average pupil will be able to work out calculations such as 46X44, 95X95 and 116X114 mentally, often faster than a calculator. Pupils will understand how to work out squares,cubes, percentages,fractions,and equations with ease. |
The opinions found on these pages are my own. They are not the opinions of the school or district where I work.
Sunday, May 15, 2011
SIP Day introduction to New Algebra curriculum (textbook)
I went to the Algebra-Geometry professional development at the high school. The district has decided to go with a traditional textbook from Pearson, "Algebra Foundations" and Geometry Foundations", that has been updated with a lot of 21st century razzle dazzle.
Karla and a few of our math teachers have taken the time to go through the entire series with a fine toothed comb. They have created a binder and web site to help teachers quickly identify the necessary content that meets current and future standards. They also created alternative Solve it activities to better introduce some of the sections.
The textbook is designed using the principals of Understanding by Design, Grant Wiggins was a consultant. Each chapter has Big Ideas and Essential Questions.
At the beginning of each chapter are seven 21st century additions.
Video introduction
Real students introduce the chapter and explain how it is applicable to the real world.
Math vocabulary
The vocabulary is recorded so students can hear the words.
Solve it
A launch problem designed to introduce each section - we may have substituted our own.
Dynamic Activities
Interactive graphs and such so student can connect Algebra to graphs.
Download
The examples are solved online or on the DVD with narration.
Online Homework
Each student can be given their assignments online.
Extra Practice
Self explanatory.
The entire textbook is online. (All teachers can and should be able to get access if you don't ask me or Karla, or another teacher for the code)
If you use the stock examples the student can replay it at home though not with your words.
Teachers can create separate classes with individual students.
Assignments can be created (well they are already created) and assigned to your virtual class.
There seem to be three standard types of assignments games - worksheets - and tutorials.
I watched one tutorial it might be a good review or supplementary lesson, the one I watched did not include the objective or a summary.
If you prefer every student in all of your classes can access the online textbook under the same user name and password.
The online textbook doesn't seem to track time spent on the assignment or give a grade when the student is finished. The student can click a button that notifies the teacher when they have finished an assignment.
The Entire textbook is on DVD. Same as above, but when your student tries to claim s/he doesn't have Internet access you can give them the DVD. (We have rights to make copies as needed) |
Logic and Utility of Mathematics: With the Best Methods of Instruction Explained and Illustrated
The Logic And Utility Of Mathematics With The Best Methods Of Instruction Explained And Illustrated. Many of the earliest books, particularly those ...Show synopsisThe Logic And Utility Of Mathematics With The Best Methods Of Instruction Explained And IllustratedHardcover reprint of the original 1850 edition-beautifully bound...Hardcover reprint of the original 1850 Logic And Utility Of Mathematics With The Best Methods Of Instruction Explained And Illustrated. Davies, Charles. Indiana: Repressed Publishing LLC, 2012. Original Publishing: The Logic And Utility Of Mathematics With The Best Methods Of Instruction Explained And Illustrated. Davies, Charles. New York, A.S. Barnes & Co.; Cincinnati, H.W. Derby & Company, 1850. Subject: Mathematics Study And Teaching.
6.
Softcover,
Pranava Books,
2013
Description:New. 390 pages. Reprinted from 1850 edition. New 2013 edition...New. 390 pages. Reprinted from 1850 400pp. First Published in 1850 It is reprint Hard Bound...New. 400pp. First Published in 1850 400 pages. ReInk Books reprint from the 1850 edition. This...New. 400 pages. ReInk Books reprint from the 1850 A.S. Barnes & co.; Cincinnati, H.W. Derby & company)
10.
Softcover,
ReInk Books,
2012-13
Description:New. 401 pages. ReInk Books reprint from the 1851 edition. This...New. 401 pages. ReInk Books reprint from the 1851 H.W. Derby & |
Video 2 in this series that will take you from your A-Level maths through to a reasonable, detailed understanding of Fast Fourier Transforms - enough to implement a Cooley-Tukey algorithm and to use other FFT implementations!
This is video 2 in the series. It introduces the continuous Fourier Transform and offers graphical explanations of how it works before proceeding to a mathematical inspection of the formula.
This series will take you from your A-Level maths through to a reasonable, detailed understanding of Fast Fourier Transforms - enough to implement a Cooley-Tukey algorithm and to use other FFT implementations!
The series looks at a general knowledge of Fourier Transforms through to conceptual understanding and then looks at the maths from the continuous Fourier Transform to the 2-Radix Cooley-Tukey algorithm for the Fast Fourier Transform.
The series uses simple examples with graphical samples to aid understanding. Videos 2 and 3 of the series look at the maths behind Fourier Transforms with some basic proofs and Videos 3 and 4 look at the discrete and fast Fourier Transforms and why they work. |
Details: A continuation of the higher-level mathematics courses or a presentation of special advanced topics in mathematics as the need and interest develops. The topics will be chosen by the instructor, generally in the area of the instructor?s specialty. Prerequisites: MATH 266, junior or senior standing and consent of the instructor. (Sp, odd years) |
INTRO TO MATHEMATICAL STATISTICS
by HOGG
New
Our Price:
$157.05
Description
Introduction to Mathematical Statistics, Seventh Edition, offers a proven approach designed to provide you with an excellent foundation in mathematical statistics. Ample examples and exercises throughout the text illustrate concepts to help you gain a solid understanding of the material. |
Obviously, for university math there should be higher prior knowledge required, and the proofs should be more formal, but I'm more worried about the writing style and how the book can catch your attention. Generally, I find the advanced math books lacking any didactic style at all. |
the Apples - simplyfluid.com
A free online learning tool that uses images of apples to help pre-schoolers
and kindergarteners learn to count up to 10 by both numbers and words. There are 4 levels of increasing difficulty for sequence and image patterns.Course Notes - J. S. Milne
Full course notes in dvi, pdf, and postscript formats for all the advanced courses J. S. Milne taught at the University of Michigan between 1986 and 1999: Group Theory; Fields and Galois Theory; Algebraic Number Theory; Class Field Theory; Modular Functions
...more>>
A Course on the Web Graph - Anthony Bonato
A comprehensive introduction to state-of-the-art research on the applications of graph theory to real-world networks such as the web graph. It is the first mathematically rigorous textbook discussing both models of the web graph and algorithms for searching
...more>>
The Cow in the Classroom - Ivars Peterson (MathLand)
Math Curse by Jon Scieszka and Lane Smith spoofs the types of word problems that educators and textbook writers invent to dress up arithmetic exercises and, supposedly, to demonstrate the relevance of math to everyday life. Canadian economist and humorist
...more>>
CP-AI-OR 2001: Third International Workshop
Third International Workshop on Integration of Artificial Intelligence and Operations Research Techniques in Constraint Programming for Combinatorial Optimization Problems. Wye College (Imperial College), Ashford, Kent UK; 8-10 April, 2001. Papers available
...more>>
CPMP-Tools - Core-Plus Mathematics Project (CPMP)
CPMP-Tools is a suite of both general purpose and custom software tools designed to support student investigation and problem solving in the 2nd edition Core-Plus Mathematics texts. The Core-Plus Mathematics Program was one of five designated as "exemplary"
...more>>
CPO Online - Cambridge Physics Outlet
A company founded by teachers and scientists that creates hands-on equipment and curriculum for teaching science, math, and technology from grades 4-12 and beyond, and provides effective professional development in science and math that is both content
...more>>
Cracking a Medieval Code - Ivars Peterson (MathTrek)
The first printed book on cryptology was written by Johannes Trithemius (1462-1516), an abbot in Spanheim, Germany, who was one of the leading intellectuals of his day. Bearing the title Polygraphiae libri sex ("Six Books of Polygraphy"), it was published
...more>>
The Crafoord Foundation
For the promotion of scientific training and research; and also the care, upbringing and education of children and young people. Apply for scientific research grants, and see grant awardees. The Royal Swedish Academy of Sciences awards a Crafoord Prize,
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CRC Press LLC
A scientific and technical publisher of books, journals, regulatory newsletters, and environmental seminars. A searchable online catalog may also be browsed by topic, among others Engineering, Computer Science, and Math & Statistics.
...more>>
Create 2D and 3D gnuplot Graphs
Input a function, vary parameters, plot 2D, 3D Surface, or 3D Contour, vary range displayed, and plot multiple functions at once. With help on various features, and links to graphing animated plots and varying the parameters. Part of the Shodor Educational
...more>>
Creative Geometry - Cathleen V. Sanders
Teachers and students will find creative and interesting "hands-on" projects for most topics in the geometry curriculum. Each project is designed to help students understand, remember, and find value in the concepts of geometry. The pages are organized
...more>>
Creative Java Puzzles - J. L. Read, Enchanted Mind
Games "good for exercising both sides of the brain." Knight's Tour: pass all the squares of the board with the knight, making only legal moves. Peg Solitaire: Remove all the pegs in the least amount of time, finishing with only one peg in the target hole.
Cross-Age Tutoring - Marah Fortson; Math Forum
An annotated bibliography: articles to help teachers deal with a classroom of learners with different abilities, different strengths and weaknesses, who are at different levels academically, behaviorally, and socially. The task of the teacher is a challenge,
...more>> |
Learning Material
The following menu user interface control may not be accessible. Tab to the next button to revert the control to an accessible version. Destroy user interface control Sign in to NCBI US National Library of Medicine National Institutes of Health
Flavors of Uncertainty: The Difference between Denial and Debate
Learning intro number theory | Ask a Mathematician / Ask a Physicist
Physicist : We occasionally get questions about free learning resources. Khan academy is excellent, and if you poke around you can find a smattering of free class notes and text books , but generally speaking the more more detailed/advanced the material, the more difficult it is to find/understand. In keeping with that tradition, here's this ↓. It was originally written for a group of " mathlete " high school students to teach them number theory and erode their egos a bit. It's Socratic , so rather than just presenting stuff to be known, it's mostly a series of leading questions. Many big names in education including MIT and Stanford offer programming courses, absolutely free.
50 Places You Can Learn to Code (for Free) Online
genetic-programming.com-Home-Page
Thinking in C CDROM | Why do you put your books on the web? | Comments from Readers | The Cover Story | Electronic Translations | Strategy | The Electronic Book | HTML Format | Making a Contribution to the Book | Downloading problems | Unzipping | Mirror Sites | Download the book | Download the source code | Win32 Compilers | Chapter 3 is a fairly intense coverage of the C that's used in C++, but if you're just getting started with all this it may be a little too intense. To remedy this, the printed book contains a CD ROM training course that gently introduces you to the C syntax that you need to understand in order to take on C++ or Java.
Bruce Eckel's MindView, Inc: Thinking in C++ 2nd Edition by Bruce Eckel
Howstuffworks
Mathtools.net - Link Exchange for the Technical Computing Community
Jazz Chord Progressions - II-V-I with 7/9/13 chords
Home » Piano Chords » Jazz Chord Progressions - II-V-I With Extensions During the past three lessons we started to talk about extensions. Now that we know how to form jazz chords I'd like us to go back to the II-V-I chord progression and see how to play it as a jazzy chord progression. We'll start with Looking at this image.
How to Design Programs
Computer Programming Algorithms Directory
If you don't know how to code, then you can learn even if you think you can't. Thousands of people have learned programming from these fine books: Learn Python The Hard Way Learn Ruby The Hard Way Learn Code The Hard Way I'm also working on a whole series of programming education books at learncodethehardway.org .
Become a Programmer, Motherfucker
The Rubik's Cube Solution
How to Solve the Rubik's Cube in Seven Steps The world's most famous puzzle, simultaneously beloved and despised for it's beautiful simple complexity, the Rubiks Cube has been frustrating gamers since Erno Rubik invented it back in 1974. Over the years many brave gamers have whole-heartedly taken up the challenge to restore a mixed Rubik's cube to it's colorful and perfect original configuration, only to find the solution lingering just out of their grasp time and time again. After spending hours and days twisting and turning the vaunted cube in vain, many resorted to removing and replacing the multi-colored facelets of the cube in a dastardly attempt to cheat the seemingly infallible logic of the cube, while others simply tossed it to the side and dubbed it impossible. The Rubik's cube, it seemed, had defeated all
Lazy Foo' Productions
Missed lectures or hate teachers? Or want to study computer science courses without going to university? … You can study anytime anywhere because there are number of free online computer science courses available on internet that are very interactive. Here is the list of 45 free online computer science courses that are designed by teaching experts from best universities of the world (almost the whole graduation!).
45 Free Online Computer Science Courses
Neural Network Tutorial
I ntroduction I have been interested in artificial intelligence and artificial life for years and I read most of the popular books printed on the subject. I developed a grasp of most of the topics yet neural networks always seemed to elude me.
Voting Methods
First published Wed Aug 3, 2011 Think back to the last time you needed to make a decision as a member of a group. This may have been when you voted for your favorite political candidate during the last election. On a smaller scale, it may have been when you took part in a committee that needed to choose the best candidate for a job or a student to receive a special award.
Real Analysis I. Professor Francis Su. Taught Spring 2010 at Harvey Mudd College. This course contains lecture video and synced transcripts. Lecture 21 could not be filmed during the original offering of the course but will be uploaded after appearing in the Spring 2011 course. If you have any questions, please don't hesitate to email us at support@learnstream.org.
Real Analysis
Linear Algebra Video Lecture Course
An Introduction to Species Counterpoint C OUNTERPOINT may be briefly defined as the art of combining independent melodies. In figure 1 the lower melody is harmonized by one a third higher - such an arrangement of the voices could be regarded as counterpoint, but there is little or no independence between the parts; for example there is no dissonance between the voices, and both rise and fall together in parallel. However, the arrangement in figure 1a demonstrates much greater independence between its two voices, with movement in one part while the other halts, divergence in the melodic direction of the parts, and with at least some occurrence of dissonance. In a longer melody the voices could peak at different times - this would also emphasize their independence.
Species Counterpoint
Hello and welcome back to my blog! This time i'm going to talk about the basic components that make up a physics engine and how to put them together; this tutorial is aimed at programmers who have a basic grasp of maths and geometry but would like to step into the world of simulation. It is my hope that if, at the beginning of this article, you are able to code the physics behind the 1972 game Pong , by the end of the article you will be equally happy writing your own constraints to use in your own physics solver! Can you code this? I've always found the title of those books of from the '…for dummies' series reassuring; after all, if a dummy can learn this stuff you should stand a good chance, right?
Physics engines for dummies
Spaun | Repository of Neural and Cognitive Models
Description: Spaun is a biologically realistic model of cognition that is not only able to perform multiple (at least 10) cognitive, perceptual, and motor tasks, but also utilizes the same model parameters across all tasks. Spaun is able to perform tasks that encompass strictly visual tasks (e.g. recognition of handwritten digits), memory tasks (e.g. forward and backward recall of a list), simple cognitive tasks (e.g. counting), and complex fluid intelligence tasks (e.g. solving the Raven's Progressive Matrices). Publication:
The Nengo Neural Simulator | Nengo
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Prime Time - Mathematicians have tried in vain to this day to discover some oreder inthe sequence of prime numbers... |
including algebra 1, algebra 2 and prealgebra algebra 1, algebra 2 and geometry algebra 1, algebra 2 and calculus |
As a complement to calculus, which is the study of continuous processes, this course focuses on the discrete, including finite sets and structures, their properties and applications. Topics will include basic set theory, infinity, graph theory, logic, counting, recursion, and functions. The course serves as an introduction not only to these and other topics but also to the methods and styles of mathematical proof.
Class Format: lecture/discussion
Requirements/Evaluation: evaluation will be based primarily on homework, classwork, and exams
Additional Info:
Additional Info2:
Prerequisites: MATH 140 (formerly 104) or MATH 130 (formerly 103) with CSCI 134 or one year of high school calculus with permission of instructor |
Numerical evaluation of the elementary functions and a large library of special
functions including the error function, the Gamma and related functions,
the exponential integral, the Riemann zeta function, Bessel and related
functions, Hypergeometric functions, statistical distributions etc. |
Mathematica Commands
Mathematica works in the interactive mode by taking input at the In[number]:= statement and then performing the requested operation and giving the result in an Out[number] statement. In Mathematica the percent sign (%) refers to the output of the previous command and Out[ number ] or % number can be used in any subsequent command as input for another command. For example, the command In[1]:= 6 results in Out[1]=6. You could then use the command In[2]:=Out[1]+5 or In[2]:=%1+5 that would result in Out[2]=11. The colon and equal sign (:= ) allow you to define variables. For example In[3]:=x:=5 would assign the value 5 to the variable x. The current value of any variable can be retrieved by typing its name at an In line. In Mathematica, functions act on arguments that are inside square brackets [] , lists are contained in curly brackets {} , and parentheses are used for grouping.
Computations and Functions
Mathematica can perform simple computations such as addition (+ ), subtraction (- ), multiplication (* or < space> , In[1]:=6 7 results in 42), division (/ ), and exponents (^ ) along with numerous built-in functions. Mathematica can perform these operations on both numerical and symbolic representations. The following examples demonstrate several of Mathematica's useful algebra and calculus functions:
Expand
expands all multiplication and power terms in an algebraic expression.
Factor
factors a polynomial over the integers. Note the use of the % sign denoting the output from the previous command.
Integrate
returns the indefinite integral with respect to the selected variable(s). The integrate command can also be used to determine a definite integral using Integrate[expr, {x, xmin, xmax}]
D
differentiates an expression with respect to the selected variable(s).
Simplify
performs algebraic manipulations to return the simplest form it can find.
N
returns approximate results using a specific number of significant digits.
Solve
solves an equation or system of equations for selected variables. Use the double equal sign (==) to define an equation.
Roots
finds the roots of the equation with respect to the selected variable.
Plotting in Mathematica
Mathematica supports numerous plot types and formatting options. To create simple x-y plots you can use the Plot command, for example:
The command Show allows you to modify the display in a graphics window. For example, the command Show[ %, Frame -> True, FrameLabel -> {"Time", "Signal"}] would add a frame and labels to the previous plot (note the % ).
These items are referred to as plot options. Plot options can also be added in the Plot command, Plot[x^2, {x,0,10}, Frame -> True]. More than one graph can be plotted on a single axis using the Plot command with a list of functions, Plot[{f1, f2, ...}, {x,xmin,xmax}]. The following commands can also be used in place of Plot for appropriate plot types: LogPlot, LogLogPlot, PolarPlot, Plot3D, ContourPlot. To determine the syntax and default options for any of the plot commands you can type ?? commandname, for example ??LogPlot.
Using Mathematica in Text Mode on Unix
Mathematica can also be run in text mode in Unix or Xwin environment. The commands demonstrated above are also valid in text mode.
Sourcing the setup files
At USC we have developed short setup files that set an appropriate path for the software you are using. You need to source these setup files in order for the software to work properly.
You can source the setup files at a UNIX prompt or add a few lines to your .login file that will source the setup files at login. To source the setup files at a UNIX prompt use the command
source /usr/usc/math/default/setup.csh
To source the setup files at login you need to add the appropriate lines to your .login file. These lines can be found in the file /usr/usc/math/default/README.USC |
Math
This program prepares students for college mathematics, and to that end, it offers six year-long courses. These courses include: Pre-algebra, Algebra 1, Geometry, Algebra 2, Pre-Calculus and Calculus. Students are required to take two years of math in the Middle School and three years of math in the Upper School. The math department determines placement of each student. |
for Technical and Vocational Students
In print for over 75 years--and continually updated to reflect the contemporary work world and the changing needs of technical/trades workers--this ...Show synopsisIn print for over 75 years--and continually updated to reflect the contemporary work world and the changing needs of technical/trades workers--this book provides an accessible, comprehensive survey of all the practical mathematical skills required on the job in industry today. Using clear, uncomplicated explanations, an abundance of illustrations, and example problems drawn from the technical and trade professions, it helps readers gain competence and confidence in a broad range of mathematical problem-solving skills--from addition of whole numbers to problems concerning threads and gearing. Features convenient-reference comprehensive tables and formulas in the back of the book. Whole Numbers. Common Fractions. Decimal Fractions. Percentage. Ratio and Proportion. Practical Algebra. Rectangles and Triangles. Regular Polygons and Circles. Solids. Metric Measure. Graphs. Measuring Instruments. Geometrical Constructions. Logarithms. Essentials of Trigonometry. Strength of Materials. Work and Power. Tapers. Speed Ratios of Pulleys and Gears. Screw Threads. Cutting Speed and Feed. Gears. A reference and tutorial on practical mathematics for anyone in the technical trades In print for over 75 years--and continually updated to...New. In print for over 75 years--and continually updated to reflect the contemporary work world and the changing needs of technical/trades workers--this book provides an accessible, comprehensive survey of all the practical mathematical skill |
Mathematics for Everyman: From Simple Numbers to the Calculus
Dispelling some of the subject's alarming aspects, this book provides, in a witty and engaging style, the fundamentals of mathematical operations. Topics include system of tens and other number systems, symbols and commands, first steps in algebra and algebraic notation, common fractions and equations, irrational numbers, much more. 1958 edition.
Unabridged republication of the edition published by Emerson Books, New York, 1958. |
This book is an introductory course on mathematical analysis, which focuses on the explanation of one of the most fundamental concepts of mathematical analysis, the concept of limits. The author is able to mix both styles relating informal language to mathematical language and giving proofs that are deep but easy to read and follow.
MSC main category:
26 Real functions
MSC category:
26A03
Other MSC categories:
26A06
Review:
"Limits, limits everywhere" is not a usual popular book on Mathematics nor a traditional analysis textbook of definitions, theorems and proofs. The author is able to mix both styles relating informal language to mathematical language and giving proofs that are deep but easy to read and follow.
The book focuses on the explanation of one of the most fundamental concepts of mathematical analysis, the concept of limits. The book's first part deals with the concepts of integer, prime, rational and real numbers, inequalities, limits, bounded sequences, and infinity series. The second part explains the special numbers e, π, and γ, infinite products, continued fractions, Cantor's different types of infinities, and the constructions of the real numbers. Throughout the book, there are some brief history remarks that explain why and for what these mathematical tools are necessary.
This is a well-written book with a style that is easy to read and follow, which can be recommended for undergraduate students interested in finding out more about Mathematics.
After his retirement as a mathematics professor at the University of Boulder (CO) in 2006, Larry Baggett wrote this memoir telling how he managed, being blinded by an unfortunate accident at the age of five, to build up a professional career as a mathematician. It is an optimistic tongue-in-cheek account of how a blind person survives in a world of the sighted. He also displays his passion for mathematics and music, of which some slightly more technical excursions are added as occasional in-text-frames.
URL for publisher, author, or book:
MSC main category:
01 History and biography
MSC category:
01A70
Other MSC categories:
00A65
Review:
Five year old Lawrence W. Baggett accidentally cut himself (he claims it is a design error of Darwinian evolution that arranges bone lengths in a human arm such that if you cut something at eye level, the knife will end up in your own eye). By sympathetic ophthalmia, he lost sight in his other eye as well, which left him totally blind apart from some vague distinction between light and dark.
That was 1944, and in those days, mainstreaming people with some bodily malfunction was not as usual as it is today. However, his mother was very determined and could convince some teachers to take her son along in the class with the other kids. Similar things happened later when he continued his education. He was always lucky to be the first blind person who was accepted in a regular program. He finally got a PhD at the University of Washington in 1966 on unitary representations of compact groups. After that he was hired by the University of Boulder where he had 12 PhD students.
With the technology available to us today, such as computers, audio-books and TeX, we can imagine a blind person writing a paper, but this was not at all obvious forty years ago. At first Braille type-writers were very primitive, and even when they improved, there was no erase button. Hence it required a lot of re-typing. Some books were available in Braille, others were later audio-recorded. For reading a paper he was depending on somebody reading it out for him. He tells us many other things that a blind person has to deal with, like traveling, finding your way in an unfamiliar city, crossing streets with heavy traffic, finding an empty seat in the audience, using a self-service counter, or the use of public toilets, and obviously how to lecture in a theater for a group of sighted students. Baggett takes us along on this journey, embedded in a sauce of gratefulness and a lot of humour. His hilarious evocation for example of what may happen when an unsighted man in a public lavatory is looking for a free toilet booth or when he bumps into the rear of a peer (or is there an 'e' missing) looking for a free urinal. Just a quote to illustrate his tongue-in-cheek phrasing "It is known to people who do research on blindness that most blind men and women can't accurately walk a straight line, which I suppose explains why so many of us get drunk-driving tickets". Until late in life he avoids being exposed as a blind person and acts as sightedly as possible. That may be the reason that he never used a guide dog (at least he never mentions one in this memoir), although he came around to using a cane.
Baggett takes us along the successive stages of his personal life and of general history (e.g. the 'revolutionary sixties', the day JFK was shot, etc.). He tells about his escapades as a student, the "girly-thing" as a teenager, how he travelled to Sweden (partly inspired by the reputation of Swedish girls), how he met his first wife, and how he later remarried with his current wife, and how he got along after being elected as head of the math department in Boulder. It is remarkable how often the reader is almost forgetting that this is a blind person telling this story. He tells about the movies he has watched, the paintings he has seen in a museum, etc. but these words get a slightly different meaning, obviously.
Besides mathematics, music has been another lifelong passion of his. He has played in several bands as a youngster and continued to do so during his later career. Both of these passions are illustrated not only by his story, but there are several framed text blocks inserted that elaborate on some of the topics that he mentions. These are not really advanced, but discuss things like the number of possibilities in a set of 6-6 domino tiles (inspired by the 163 possibilities formed by a 6 by 2 dot matrix of the Braille alphabet), or the comma of Pythagoras, the sequence of musical chords, the limit of a sequence of numbers, the irrationality of √2, mathematical induction, etc. Sometimes he just challenges the reader giving a question of an IQ test he had to do: what is the next number in the sequence 6, 42, 7, 12, 48, 16, 18.
This account is indeed 'on the sunny side' (and this probably refers to the jazz standard song `On the sunny side of the street' performed by many of his much admired jazz heros). Apart from his accident as a child, Baggett may have been lucky at many other instances of his life, nevertheless this book testifies that, even 'in the dark' it is possible by using one's creativity and perseverance to achieve remarkable things in life.
This is a revised edition of previous version (2010) with adaptations for the new release of Python 3.0 (2008). It treads the classical subjects of an introductory course on numerical methods for engineering students with chunks of Python code implementing the algorithms. Python is just a working tool subordinate to the numerical techniques. So this should not be mistaken as a programming or a Python course.
URL for publisher, author, or book:
MSC main category:
65 Numerical analysis
MSC category:
65-01
Other MSC categories:
65–04,
Review:
Python is a general purpose programming language supporting object-oriented and structured programming. It is free, simple to use and implement, and well structured, and equally useful for non-numerical as for numerical applications. For these reasons it is sometimes chosen as a language for a first programming course. It is clear that in such case, a first course on numerical methods prefers using Python as a tool for implementing and testing the algorithms over an alternative such as MATLAB, or scilab, the open source MATLAB clone.
The present book is written in line with the previous observations. It is assumed that the student/reader is familiar with some programming language that is preferably, but not necessarily, Python. For those who do not know Python yet, a short introduction is given, and it as assumed that their programming skills suffice to learn the details along the way. Thus the book should not be considered as a course on Python. Only part of the language is used. For example, the only object that is used is an array. On the other hand, modules with many predefined functionality beyond the Python core such as numpy and matplotlib.pyplot are essential. Source code for the algorithms in the book are available for download at the CUP website.
After the introductory chapter on Python, the book follows the classical structure and items of a first numerical analysis course: Solution of linear systems, interpolation and curve fitting, roots of equations, numerical differentiation and integration, (ordinary) differential equations (initial value and boundary value problems), (symmetric) eigenvalue problems and (elementary) optimization. It is a true hands-on course where formal proofs are almost always omitted or just replaced by some motivating examples. There is an ample number of exercises, mostly numerical (i.e., not asking for a proof or a generalization or the derivation of a formula) and they are often taken from engineering applications.
Some peculiarities that struck me are the following.
+ There is no list of references. With proofs and more formal aspects missing, some students may be interested in further reading or a more advanced approach. They are left on their own, although these references hould be readily available in any library or even on the Internet.
+ There is no treatment of the mechanism of rounding errors in floating point computations and neither is error analysis or the numerical stability of a method formally included (except for stability and stiffness of ODE solvers). Remarks on rounding errors, and hence on stability, are downgraded to remarks sometimes with computing in double precision as a possible cure.
+ Polynomial interpolation is included in extension, even rational and spline interpolation are discussed, but not the choice of the interpolation points like adaptive interpolation or the use of Chebyshev points for a finite interval. A simple idea that may influence the error and the convergence considerably.
+ Speed of convergence for iterative methods is mentioned but not formally discussed.
+ Multivariate optimization is very concise as compared to the extensive discussion of less important issues. Discussion includes only a method of M. Powell and the simplex method (not to be confused with the simplex method of linear programming which is not included).
So we may conclude that this is a practical introduction, pushing the theory as far in the background as possible. There is a lot of material, probably far too much for a single course. Some sections that could be deleted for a shorter course are marked with an asterisk.
For those who are familiar with the second edition, the changes (apart from the necessary adaptation of the code to Python 3.0) include: an introduction to matplotlib.pyplot in chapter 1; an interpolating polynomial plot routine is added to the interpolation chapter; in the chapter on differential equations, the Taylor series method is dropped in favor of Euler's method; an improved implementation is given for the Jacobi method for eigenvalues as well as for the Runge-Kutta method for differential equations; and some new problems are added.
Mainly relying on two-dimensional transformations, the book is a mixture of proper mathematics explaining some aspects of Euclidean and non-Euclidean geometry, philosophical discussions on aesthetics, and all of this amply illustrated with many examples from mainly two-dimensional pieces of art, and one chapter on Bach's music. These examples illustrate many forms of symmetry, tessellations, hyperbolic and projective geometry, the use of perspective, other non Euclidean geometries, etc.
URL for publisher, author, or book:
MSC main category:
00 General
MSC category:
00A66
Other MSC categories:
51M20, 51N10, 51N15, 51N20, 51N25
Review:
The cover states that this book grew out of a liberal arts course. Felipe Cucker is Chair Professor of Mathematics at the City University of Hong Kong. The result is a collection of chapters, some of which are just plain mathematics, others analyse the philosophical and psychological aspects of aesthetics, and many discuss a wide diversity of works of art and how the mathematics are recognized in their features (like symmetry or translation invariance for example) or how mathematics have influenced the techniques available to the artists (like perspective or hyperbolic geometry).
Almost all examples are two-dimensional, which is related to the mathematics that are covered. These are all geometric, treating transformations in the plane or different kind of projections and non-Euclidean geometry. So most art examples are graphical like paintings and drawings but also carpets, and occasionally poetry and dancing is mentioned. One exception is a complete chapter devoted to Bach's canons, but architecture and sculpture, which is clearly three-dimensional, is almost completely absent.
After an appetizing introduction showing symmetry and structure in work by Simone Martini (painting), John Milton (poetry) and Johan Sebastian Bach (music), the first chapter introduces geometry and its history from Euclid to Descartes and this is followed by a mathematical treatment of plane transforms: translation, rotation, reflection, glide, isometry, completely with definitions and proofs. Artistic examples illustrate the mathematics in another chapter where it is shown that there are exactly 7 friezes (translation invariant pattern in one direction) and 17 wallpapers (translation invariant pattern for two independent vectors). Pieces of art with planar symmetry are easily found. Tessellations and patterns in carpets from Central Asia, Chinese lattices and of course Escher's work.
Much more philosophical is an analysis of George D. Birkhoff's attempt to define a measure of aesthetics and Ernst H. Gombrich's sense of order. We often see symmetry when it is not really there. Da Vinci's Vitruvian man is not perfectly symmetric. A sense of beauty is raised by a balance disorder and boredom. This is illustrated with several examples from op-art, and for example repetitive work by Andy Warhol, but also from ballet performances and the rhyme and rhythm of poetry. Mathematics re-enter with homothecies, similarities, shears, strains and affinities and conics, which is illustrated by the use of the ellipse (a circle in perspective) in the Renaissance. More patterns are illustrated with musical canons, in particular the ones in J.S. Bach's Musical Offering.
The introduction of perspective in European paintings triggers some more mathematics introducing projective geometry and projections. This allows to produce proper representation of a reflection in a sphere, but also optical illusions based on false perspective. The rules of perspective are left with the start of cubism and modern art. The parallel in mathematics is that Euclidean geometry is left to introduce alternatives based on axiomatic systems and formal languages. In a final short chapter, Cucker ponders briefly on the geometry to describe our universe, but this requires to leave the two-dimensional world he has been discussing so far.
Rule-driven creation is moved to an appendix. Literature does not have the same geometrical basis as the other examples according to Cucker, yet he describes some patterns of constrained writing like anagrams, palindromes and other word plays.
This wonderful survey shows that, even though the author has restricted his approach mainly to two-dimensional geometry and transformations of the plane, it should be clear by now that this is still a very broad area when this is related to visual (and aural) art. From Euclidean geometry to Gödel's completeness theorem, from stone age artifacts to modern dance theater, from short biographies to quantitative aesthetics, the scope is enormous, forcing the author to be selective. The illustrations used are not always the ones that are best known though. So there is certainly something new to be discovered for every reader. The book grew out of a course, and so it is obviously possible to extract some interesting lectures from the material that is presented. |
Mathematics Syllabus
Engineering Entrance
,
Punjab CET
Maths Syllabus Punjab Technical University Common Entrance Test:
NUMBER SYSTEM
Statements of algebraic and order properties of the system of natural numbers, integers, rational numbers and real numbers and simple basic deductions from theses properties.
Complex numbers. Representation of complex numbers as points in a plane.Algebra of complex numbers. Real and imaginary parts. Modulus and argument of a complex number. Conjugate of a complex number, cube roots of unity.
Statements of the principal of mathematical induction in respect of natural numbers and simple applications.
CO-ORDINATE GEOMETRY
Distance formula and section formula.
Equation of line in a plane. General equation of first degree. Angle between two lines. Parallel and perpendicular lines. Distance of a point from a line. Family of lines.
Equation of a circle. General equation. Equation of tangent and normal to a circle. Radical axis of two circles. Family of circles.
Position Vector of a point. Section formula. Application of Vector to some geometrical results.
Scalar and Vector product of two vectors.
Scalar triple product, vector triple product.
THREE DIMENSIONAL GEOMETRY
Decomposition of a vector into three non-coplanar directions i, j, k as base in 3-dimensons.
Angle between two vectors.
Distance between two points. Section formula.
Equations of lines and planes in 3-D. Angle between two lines, between a line and a plane as also between two planes. Distance of a point from a line and from a plane. Shortest distance between two lines. Equation of any plane passing through the intersection of two planes.
Equation of a sphere in the form (r-C)2=a2, Equation of a sphere with the position Vector
Angles and their measure in degrees and radians. Trigonometric functions of angles of arbitrary magnitude. Addition formulae. Sine, cosine, and tangent of multiples and sub multiples of angles. Periodicity and graph of sine, cosine and tangent functions.
Trigonometrical ratios of related angles.
Solutions of simple trigonometric equations.
Sine and cosine formulae for triangles and simple cases of solutions of triangles, problems on heights and distances.
Inverse trigonometric functions.
QUADRATIC EQUATIONS
Quadratic equations and their solutions. Relationship between the roots and the coefficients. Formation of quadratic equations with given roots. Criteria for the nature of the roots of quadratic equation.
Determinants: minors and cofactors. Expansion of determinant, properties of elementary transformation of determinants.
Application of determinants in solutions of equations. Cramer's rule.
Adjoint and inverse of a matrix and its properties. Applications of matrices in solving simultaneous equations in three variables.
DIFFERENTIAL CALCULUS
Concept of real function, its domain and range, one-one and inverse functions, composition of functions. Notions of right hand and left hand limits and the limits of a function. Fundamental theorems on limits.
Derivative of a function, its geometrical and physical significance, relationship between continuity and differentiability.
Derivative of sum, difference, product, quotient function and of the functions of a function(chain rule), derivatives of trigonometric functions. Logarithmic and exponential functions. Differentiation of functions expressed in parametric form, derivatives of higher order.
Order and degree, formation of differential equation, general and particular solution, solution by the method of variables, separable. Homogeneous equations and their solutions. Solution of the linear equation of the first order with constant coefficients.
Integration as the inverse of differentiation, indefinite integral, properties of integrals, fundamental integrals involving algebraic trigonometric and exponential functions, integration by substitution and by parts. Definition of definite integrals as the limit of a sum illustrated by simple examples, fundamental theorem of Calculus, evaluation of definite integrals, transformation of definite integrals by substitution, properties of definite integrals.
Application to determination of area under plane curves in simple cases.
STATISTICS
Population and sample, Measures of central tendency and dispersion, Point and internal estimation(of mean only), Scatter diagrams and Pearson's correlation coefficients. Calculations of the regression coefficients and the two lines of regression by the method of least squares. |
Undergraduate Program:
Senior Project Goals
In order to help a student choose a senior project, it
is important for the faculty member to keep two things in mind: (1) the
goal of the project, as opposed to the variety of goals in mathematics
courses; (2) the level of difficulty of the project, which should be appropriate
for the student's mathematical background. My view is that the primary
goal of the senior project is to provide the student an opportunity to
create a coherent story involving several proofs or calculations, each
of which is at a level that the student has already shown he or she can
handle.
It is not reasonable to ask a student who has typically
earned B's or B-'s on tests to now create a coherent story involving the
types of problems that on tests are used to decide who deserves an A or
A-. Rather the goal is a to provide a capstone experience in which the
student combines mathematics that he or she already understands with some
new mathematics or applications at the same level. |
Quick Links
Learning Goals
These learning goals were created by a working group of faculty -- both those in physics education research and those
in other areas of research. This list represents what we want students to be able to do at the end of the
course (as opposed to what content is expected to be covered, as in a syllabus).
Math/physics connection: Students should be able to translate a physical description of a junior-level
electromagnetism problem to a mathematical equation necessary to solve it. Students should be able to explain the
physical meaning of the formal and/or mathematical formulation of and/or solution to a junior-level electromagnetism
problem. Students should be able to achieve physical insight through the mathematics of a problem.
Visualize the problem: Students should be able to sketch the physical parameters of a problem (e.g., E
or B field, distribution of charges, polarization), as appropriate for a particular problem.
Organized knowledge: Students should be able to articulate the big ideas from each chapter, section,
and/or lecture, thus indicating that they have organized their content knowledge. They should be able to filter this
knowledge to access the information that they need to apply to a particular physical problem, and make connections/links
between different concepts.
Communication. Students should be able to justify and explain their thinking and/or approach to a
problem or physical situation, in either written or oral form.
Problem-solving techniques: Students should be able to choose and apply the problem-solving technique
that is appropriate to a particular problem. This indicates that they have learned the essential features of different
problem-solving techniques (eg., separation of variables, method of images, direct integration). They should be able to
apply these problem-solving approaches to novel contexts (i.e., to solve problems which do not map directly to those in
the book), indicating that they understand the essential features of the technique rather than just the mechanics of its
application. They should be able to justify their approach for solving a particular problem.
Approximations: Students should be able to recognize when approximations are useful, and use them
effectively (eg., when the observer is very far away from or very close to the source). Students should be able to
indicate how many terms of a series solution must be retained to obtain a solution of a given order.
Series Expansions: Students should be able to recognize when a series expansion is appropriate to
approximate a solution, and complete a Taylor Series to two terms.
Symmetries: Students should be able to recognize symmetries and be able to take advantage of them
in order to choose the appropriate method for solving a problem (eg., when to use Gauss' Law, when to use
separation of variables in a particular coordinate system).
Integration: Given a physical situation, students should be able to write down the required partial
differential equation, or line, surface or volume integral, and correctly calculate the answer.
Superposition: Students should recognize that – in a linear system – the solutions may be formed
by superposition of components.
Problem-solving strategy: Students should be able to draw upon an organized set of content knowledge
(LG#3), and apply problem-solving techniques (LG#4) to that knowledge in order to organize and carry out long analyses
of physical problems. They should be able to connect the pieces of a problem to reach the final solution. They should
recognize that wrong turns are valuable in learning the material, be able to recover from their mistakes, and persist in
working to the solution even though they don't necessarily see the path to the solution when they begin the problem.
Students should be able to articulate what it is that needs to be solved in a particular problem and know when they have
solved it.
Expecting and checking solution: When appropriate for a given problem, students should be able to
articulate their expectations for the solution to a problem, such as direction of the field, dependence on coordinate
variables, and behavior at large distances. For all problems, students should be able to justify the reasonableness of
a solution they have reached, by methods such as checking the symmetry of the solution, looking at limiting or special
cases, relating to cases with known solutions, checking units, dimensional analysis, and/or checking the scale/order of
magnitude of the answer.
Intellectual maturity: Students should accept responsibility for their own learning. They should be
aware of what they do and don't understand about physical phenomena and classes of problem. This is evidenced by asking
sophisticated, specific questions; being able to articulate where in a problem they experienced difficulty; and take
action to move beyond that difficulty.
Maxwell's Equations: Students should see the various laws in the course as part of the coherent field
theory of electromagnetism; ie., Maxwell's equations.
Build on Earlier Material. Students should deepen their understanding of Phys 1120 (Freshman
Electricity and Magnetism) material. I.e., the course should build on earlier material.
These learning goals represent our specific topical learning goals organized by chapter in Griffith's text. While they
were confirmed by faculty, less consensus was reached regarding these topic scale goals.
Chapter 2: Electrostatics
State Gauss' Law and construct the 3 Gaussian surfaces (sphere, cylinder, pillbox)
Use Cartesian, spherical and cylindrical coordinates appropriately when constructing integrals and surface and
volume elements
Goals - Students should be able to,
Electric Field
Students should be able to state Coulomb's Law and use it to solve for E above a line of charge, a loop of
charge, and a circular disk of charge.
Students should be able to solve surface and line integrals in curvilinear coordinates (when given the
appropriate formulas, as in the inner-front cover of Griffiths).
Divergence and Curl of E; Gauss' Law
Students should recognize when Gauss' Law is the appropriate way to solve a problem (by recognizing
cases of symmetry; and by recognizing limiting cases, such as being very close to a charged body).
Students should be able to recognize that E comes out of the Gaussian integral only if it is constant along
the Gaussian surface.
Students should be able to recognize Gauss' Law in differential form and use it to solve for the charge
density given an electric field E.
Electric Potential
Students should be able to state two ways of calculating the potential (via the charge distribution and
via the E-field); indicate which is the appropriate formulation in different situations; and correctly
evaluate it via the chosen formulation.
Students should be able to calculate the electric field of a charge configuration or region of space when
given its potential.
Students should be able to state that potential is force per unit charge, and give a conceptual description
of V and its relationship to energy.
Students should be able to explain why we can define a vector potential V (because the curl of E is zero and
E is a conservative field).
Students should be able to defend the choice of a suitable reference point for evaluating V (generally
infinity or zero), and explain why we have the freedom to choose this reference point (because V is arbitrary
with respect to a scalar – its slope is important, not its absolute value).
Work and Energy
Students should be able to calculate the energy stored in a continuous charge distribution when given the
appropriate formula
Students should be able to explain in words what this energy represents.
Conductors
Students should be able to sketch the induced charge distribution on a conductor placed in an electric field.
Students should be able to explain what happens to a conductor when it is placed in an electric field, and
sketch the E field inside and outside a conducting sphere placed in an electric field.
Students should be able to explain how conductors shield electric fields, and describe the consequences of
this fact in particular physical problems (e.g., conductors with cavities).
Students should be able to state that conductors are equipotentials, that E=0 inside a conductor, that E is
perpendicular to the surface of a conductor (just outside the conductor), and that all charge resides on the
surface of a conductor.
Maxwell's Equations
Students should be able to interpret the first and second Maxwell's equations for electrostatics (divergence
and curl of E) and use them to describe electrostatics (i.e., Gauss' Law is just one application of the first
law).
Recognize the wave equation in Cartesian coordinates, and state that e^(ikx) is a solution
Recognize the solution to separation of variables in Cartesian coordinates.
Recognize that a function can be expanded in terms of a complete basis, such as sin and cos.
State that conductors are equipotentials.
Goals - Students should be able to,
Laplace's Equation
Students should recognize that the solution to Laplace's equation is unique.
Method of Images
Students should realize when the method of images is applicable and be able to solve simple cases.
Students should be able to explain the difference between the physical situation (surface charges) and the
mathematical setup (image charges).
Separation of Variables/Boundary Value problems
Students should be able to state the appropriate boundary conditions on V in electrostatics and be able to
derive them from Maxwell's equations.
Students should recognize where separation of variables is applicable and what coordinate system is
appropriate to separate in.
Students should be able to outline the general steps necessary for solving a problem using separation of
variables.
Students should be able to state what the basis sets are for separation of variables in Cartesian and spherical
coordinates (ie., exponentials, sin/cos, and Legendre polynomials.)
Students should be able to apply the physics and symmetry of a problem to state appropriate boundary conditions.
Students should be able to solve for the coefficients in the series solution for V, by expanding the potential
or charge distribution in terms of special functions and using the completeness/orthogonality of the special
functions, and express the final answer as a sum over these coefficients.
Multipole Expansion
Students should be able to explain when and why approximate potentials are useful.
Students should be able to identify and calculate the lowest-order term in the multipole expansion (i.e., the
first non-zero term).
Students should be able to sketch the direction and calculate the dipole moment of a given charge
distribution.
Students should be able to calculate current density J given the current I, and know the units for each.
Students should be able to explain, in words, what the charge continuity equation means.
Students should be able to state the vector form of Ohm's Law and when it applies.
Students should be able to calculate the current I, K and J in terms of the velocity of the particle or in
terms of each other.
Magnetic Fields and Forces
Students should be able to describe the trajectory of a charged particle in a given magnetic field.
Students should be able to sketch the B field around a current distribution, and explain why any components
of the field are zero.
Students should be able to explain why the magnetic field does no work using concepts and mathematics from
this course.
Biot-Savart
Students should be able to state when the Biot-Savart Law applies (magnetostatics; steady currents, dp/dt=0).
Students should be able to compare similarities and differences between the Biot-Savart law and Coulomb's Law.
Students should be able to choose when to use Biot-Savart Law versus Ampere's Law to calculate B fields, and
to complete the calculation in simple cases.
Divergence and Curl of B
Students should be able to draw appropriate Amperian loops for the cases in which symmetry allows for
solution of the B field using Ampere's Law (ie., infinite wire, infinite plane, infinite solenoid, toroids),
and calculate Ienc.
Students should be able to make comparisons between E and B in Maxwell's equations
Magnetic Vector Potential
Students should be able to explain why the potential A is a vector for magnetostatics, whereas it's a scalar
(V) in electrostatics. Ie., that the source of magnetic fields (the current) is a vector, whereas the source
of electric fields (charge) is not.
Students should recognize that A does not have a physical interpretation similar to V, but be able to
identify when it is useful for solving problems.
Separation of Variables/Boundary Value Problems
Students should be able to state the appropriate boundary conditions on B in magnetostatics and be able
to derive them from Maxwell's equations
Maxwell's Equations
1. Students should be able to interpret the third and fourth Maxwell's equations for electrostatics
(divergence and curl of B) and use them to describe magnetostatics (i.e., Ampere's Law and Biot-Savart law
are just applications of these laws).
Chapter 7: Electrodynamics
There is not a general consensus on whether this chapter should be covered in the first semester. Most students from this
course go on to take the second semester and will see Maxwell's equations there. Even if this material is covered in here,
it may still be prudent to review at the beginning of the following semester. |
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Plane Geometry
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6. Fundamental theorem of algebra. Sketch of a proof
using the notion of loops and their winding numbers.
7. Extensions of the field of rational numbers.
Algebraic and transcendental numbers. Gauss lemma. Eisenstein's
criterion of ireducibility of a polynomial. Gauss field and Gauss
integers. Algebraic integers. Quadratic extensions. Constructibility
by ruler and compass.
8. Galois theory. Splitting fields. Automorphisms of
a field. Galois group of a field exten-sion. Galois group of a
polynomial. Separable polynomials and separable extensions. Basic
theorems of Galois theory. Normal extensions. Fundamental theorem of
Galois theory.
Teaching methods and student assessment.
Attending lectures and exercises is compulsory. The examination
consists of both written and oral part after completing lectures and
exercises. During the semester students can take tests which replace
the written part of the examination.
Teaching methods and student assessment.
Attendance is mandatory in terms of both lectures and exercises. The
final assessment consists of the written and oral examination, which
can be taken upon completion of lectures and exercises. During the
semester students can take two tests, which then replace the written
examination.
Course objective. This course will introduce
students to fundamental fields of financial analysis, with the
emphasis placed on the methodological and instrumental aspect and
the application of mathematical and statistical methods and models.
Teaching methods and student assessment.
Lectures and seminars (mostly by using computers). Students'
knowledge referring to the application of various methods is
continuously assessed through tests and projects. The final
examination consists of the written and the oral examination.
Course objective. The aim of the course is to
qualify each student for the application of statistical methods to
solving economical problems on a micro and a macro level. The
emphasis is put on understanding the context of business decisions
making. Students are required to understand statistical methods. The
course is organised such that fundamentals are made up of
consideration of an economical problem, after which there follows
application of statistical methodology aiming at analysis and
development of a solution to the given problem.
Prerequisites. Probability, Statistics.
Course contents. Application of statistical methods in solving certain financial
problems. Analysing outstanding accounts (invoices) and charging for
those accounts. Testing the influence of different factors on equity
price. Control of money supply. Credit rate testing. Analysing IBM's
equity prices.
Application of statistical methods in production control process:
Testing product quality. Testing the production process. Testing
production quality. Anticipating the number of working hours
required for finishing the work. Modelling (forming) the expected
duration of a machine.
Application of statistical methods in marketing: Promotion
efficiency in product sale. Influence of different ways of product
advertising. Anticipating the market value of a product. Choosing
hotel location. Testing the influence of media on public opinion.
Modelling the product sale per month. Testing the influence on
product sale. Testing the influence of advertising on profit rate.
Application of statistical methods in management: Testing the
efficiency of training programmes for managers. Comparison of
managers' efficiency in making business. Testing the efficiency of
sale strategies.
Application of statistical methods in planning: Planning future
constructive (structural) projects. Planning construction of a new
plant. Planning the retail trade based on a month data analysis.
Application of statistical methods in certain macroeconomic problems:
Testing the difference between unemployment rates in different areas.
Analysing the index of consumer goods as part of GDP. Monitoring the
month prices of gold on the world market. Product consumption in
relation to average retail prices in a certain period of time.
Relation between month salary and productivity. Phillips curve. Cobb-Douglas
curve of production. Analysing personal goods and personal income.
Forming product demand. Analysing work force considering age, gender,
education, unemployment rate and its changes. Predicting equity
prices. Demand and supply function. Company's investing attitude.
Analysing product spending per capita. Analysing an increase in
country's' revenue. Modelling the average spending as a function of
average revenue.
Teaching methods and student assessment.
Teaching methods used in this course are: lectures, exercises, group
discussions and case studies. Attendance and activity are required.
Continuous student assessment is carried out through tests and
assignments during the semester. In addition, each student has to
develop a project assignment consisting of the data analysis worked
out on one actual economical problem. For that purpose, students
will use statistical programmes SAS and Statistica (StatSoft). The
final examination consists of the written and the oral part.
Course objective. The objective of this course
is to make students familiar with the fundamental concepts and
results of time series analysis. Students will be introduced to
classical and modern methods in modelling real-life time series.
Special attention will be dedicated to the applications of time
series models in economics and financial mathematics. In the
practical part of the course students are supposed to learn
necessary techniques and apply them on computers.
Prerequisites. Probability. Statistics. Random processes.
Teaching methods and student assessment.
Classes will be carried out in a series of lectures, exercises and
practical classes at the computer lab. Grading will be based on the
final examination (oral or written), assignments during the semester
and/or project work.
Course objective. Students should learn about
different data models, principles of database modelling, and acquire
skills required to use databases. Through exercises they should gain
an insight into database management systems (MS Access, MS SQL and
MySQL) and basic database administration.
Computational geometry is a recent field of computer
science, originating around 1978, that studies geometric problems
from an algorithmic or complexity-theoretic point of view. Results
of interest in this field include efficient algorithms and lower
bounds for the following example classes of problems: Content:
1. Perform a geometric construction, e.g., compute the convex hull
or Voronoi diagram of a given set of points.
2. Generate additional geometric structure, e.g., triangulate or
otherwise mesh a given set of points or polygon.
3. Extract geometric information from given data, e.g., find the
closest pair of points in a given set of points.
4. Preprocess input into a data structure to support fast queries,
e.g., range queries or point location.
5. Maintain information about data subject to a sequence of updates
and queries, e.g., collision detection. Computational geometry is
also closely tied to discrete geometry, which is more broadly
interested in determining properties of geometric figures and
operations, even if they do not immediately lead to algorithms (although
frequently they do).
Computational geometry is also closely tied to discrete geometry,
which is more broadly interested in determining properties of
geometric figures and operations, even if they do not immediately
lead to algorithms (although frequently they do).
Expected work: Throught the semester several
projects will be assigned, some involving programming skills. The
projects are normally undertaken by individuals. Each project is
followed by a class presentation.
Course objective. Make students familiar with
basic cognitions in didactics referring to organisation of teaching
in primary school, develop skills necessary for application,
realisation and evaluation of the teaching process. Become familiar
with the development of didactical thoughts. Understand basic
methods of research in didactics. Develop critical opinion towards
the application of methods, forms and ways of teaching. Become
familiar with the structure and importance of curriculum;
preparation, realisation and evaluation of teaching; teaching as
communication; teaching systems – from a theoretical and a practical
viewpoint. Usage and a critical approach to teaching technologies.
Educational pluralism – alternative schools.
Course objective. At the introductory level
students should be introduced to fundamental ideas and methods of
mathematical analysis, which represent the basis for many other
courses. During lectures basic terminology would be explained in an
informal way, their utility and applications would be illustrated.
During exercises students should master an adequate technique and
become trained for solving concrete problems. The programme is the
same for all branches.
Prerequisites. High-school knowledge.
Course contents.
1. Introduction. Real numbers, supremum and infimum of a set,
absolute value, intervals. Complex numbers.
2. Functions. Definition of a function, representation and basic
properties of functions. Composition of functions and the inverse
function. Elementary functions.
3. Sequences. Definition of a sequence. Some special sequences.
Convergent sequence. The number e.
4. Limit and continuity of a function. Limit of function. Properties
of limit. One-sided limits. Infinite limits and limits at infinity.
Asymptotes. Continuous functions.
5. Differential calculus. Tangent line and velocity problems.
Concept of derivative. Differentiation rules and derivatives of
elementary functions. Implicit differentiation. Parametric
differentiation. Lagrange's mean-value theorem. Higher-order
derivatives. Taylor's theorem.
6. Applications of the derivatives. Differentials. Newton's method
of tangents. L'Hospital's rule. Applications of the derivatives (tangent
and normal, increase and decrease of a function, local extrema,
convexity and concavity of a graph, points of inflection, sketching
the graph of a function, curvature of a curve The goal of this course is to
make students familiar with the structure of their graduation theses
(contents, introduction, body of the text, literature, summary) as
well as appropriate graphic design (page layout, typeface, reference
and citation, formulae, diagrams, figures, tables, etc.). Prior to
the thesis defense, every student is obliged to present his/her
thesis to second-year Master level students of mathematics and fifth-year
Master level students obtaining a degree in mathematics and computer
education.
Teaching methods and student assessment.
Students' active participation in the seminar is required. Their
regular attendance is confirmed by the signature of the seminar
lecturer.
e-Business I002 (2+0+2) - 4
ECTS credits
Course objective. To make students familiar
with changes in business influenced by information and Internet
technology. Basic terms, principles and models important for e-business
will be described during the course lectures. The case study method
will also be used by analysing the advantages and limitations of e-business
strategies in companies that use B2C and B2B models. Students will
be able to analyse assigned cases and propose an e-business strategy
of a company through project assignments. The course will provide an
insight into possibilities of Internet and information technology in
business, as well as its impact to business successfulness.
Teaching methods and student assessment.
Lectures and seminars are obligatory. Student knowledge is examined
during the semester through homework and project assignments. There
is also a final examination (written and oral).
Course objective. The objective of this course
is to systematise, consolidate and deepen the knowledge of the
elementary primary-school geometry, without giving axiomatic of
geometries. Classical geometrical contents will be updated by
demonstrations on computers.
Prerequisites. Not necessary.
Course contents.
1. Introduction to the planimetry. Basic objects of geometry in
plane (points and straight lines). Axioms of Euclidean geometry
plane. Axioms about paralleles. (The axioms will be given only as
information and dealt with very elementary.)
2. Prominent sets of points in the plane. Half-line. Segment. Convex
sets in the plane. Half-plane. Angle. Measure of angle. Vertical
angles. Angles with parallel arms and angles with perpendicular arms.
Angles along transversal. Triangle. Sum of angles in a triangle.
Relation of triangle. Quadrangle. Diagonal of a quadrangle.
Trapezoid. Parallelogram. Rhomb. Rectangle. Square. Quadrangles with
perpendicular diagonals. Multiangles. Circumference and circle. (Only
proofs referring to angles will be dealt with in detail; all other
concepts will be only defined.)
3. Congruence of a triangle. Definition of triangle congruence.
Triangle congruence theorems. Perpendicular bisector theorem. Four
basic constructions of a triangle. Characterisation of a
parallelogram and a rhomb. The midline of a triangle theorem. Four
characteristic points of a triangle. Circumcircle and incircle of a
triangle. The midline of a trapezoid theorem. Theorem about the
bisector of an angle.
4. Perimeter and area. Perimeter and area of a polygon. Areas of
square, parallelogram, triangle, trapezoid, quadrangle with
perpendicular diagonals. Heron's formula. Connection between the
area of a triangle and its sides and the radius of its escribed
circles. Area of a circle. Length of a circumference.
5. Similarity of triangles. Thales' theorem of proportion. Theorem
about bisector of an interior angle in a triangle. Definition of
similarity of triangles. Pythagorean theorem (some proofs) and its
converse. Euclidean theorem.
6. Theorems about circumference. Theorem about peripherical and
central angle. Thales' theorem about angle on a diameter.
Circumscribed and inscribed quadrilateral.
7. Plane mapping. Isometries of a plane. Axial and central symmetry.
Rotation. Translation. Homothety. Eulerean line. Mapping of
similarity.
8. Introduction to stereometry. Basic objects of geometry of space
(points, lines and planes). Axioms of Euclidean geometry of space.
Determination of plane and a line in the space. Halfspace. Parallel
lines and planes. Perpendicular lines and planes. Theorem of three
normals.
9. Angles between lines and planes. Angle between two lines. Angle
between line and plane. Angle between two planes.
10. Distance in the space. Distance from point to plane. Distance
from point to line. The shortest distance between skew lines.
Symmetral planes of a segment and of a couple of planes. Dihedrons
and trihedrons.
11. Polyhedra. Idea of polyhedron. Some kinds of polyhedra (pyramid,
bipyramid, prism). Eulerean formula for polyhedra. Regular polyhedra
(Platonean bodies). Volume and surface area of a polyhedron -
rectangular parallelepiped, parallelepiped, prism, pyramid and
truncated pyramid. Cavalieri's principle.
12. Round bodies. Cylindar. Cone. Sphere. Volume and surface of
round bodies - volume and surface area of cylinder, cone, sphere.
Teaching methods and student assessment. Students are obliged to be
present in classes and participate actively in the practical part.
Students' knowledge is assessed during the semester through tests
and homework. Written part of the final examination can be replaced
by tests.
Course objective. To refresh and broaden
students' knowledge of elementary mathematics, which is necessary as
a strong base for further study. Educational contents of this course
are equal for all study branches and an oral part takes places upon completion of lectures
and exercises.
Course objective. To refresh and broaden
students' knowledge of elementary mathematics, which is necessary as
a strong base for further study. Educational contents of this course
are equal for all study branches.
Prerequisites. Not required.
Course contents.
1. Axioms of Euclidean geometry in the plane.
2. Isometric mapping (axial symmetry, rotation, central symmetry,
translation). Homothety and similarity. Angles (degrees, radians).
3. Geometry of triangle (congruence; center of gravity of triangle,
orthocenter, center of circumcircle and of incircle; similarity of
triangles; right triangle).
4. Area and perimeter of polygons and other geometrical figures in
the plane.
5. Applications of trigonometry (sine and cosine theorem for
triangle, trigonometric equalities in a right triangle, in other
geometrical figures).
6. Stereometry – volumes and surface areas of some geometric solids
in the space and an oral part takes places upon completion of lectures
and exercises.
Course objective. Students should acquire
fundamental terminology from the fields of mathematics and computer
science as well as apply structures typical of ESP (English for
Specific Purposes). They should be taught and trained how to read
various pieces of literature from the fields of mathematics and
computer science as well as to carry out conversation referring to
some basic topics ESP (English for Specific Purposes).
They should be taught and trained how to read and understand various
pieces of literature pertaining to mathematics and computer science,
discuss topics in their fields of study and translate simple ESP
texts from Croatian into English. Students should also be taught how
to individually present a selected topic in Englishshould individually present a selected ESP topic, which altogether
affects their final grade. Students' knowledge is continuously
assessed by four tests, two per each semester, and the oral part of
the examination takes place at the end of academic year.
Course objective. Students will learn basic
concepts (terms, idea, knowledge), symbols and principles of
mathematics of finance and actuarial mathematics through lectures,
tutorials and special assignments.
Teaching methods and student assessment.
Students are obliged to attend lectures and exercises. Exercises are
auditory and laboratory, at which students use Mathematica and/or
MatLab software packages. Students may take an examination only
after having attended all lectures and exercises. The final
examination consists of the written and the oral part. Students may
take tests during the semester which replace the written part of the
final examination. Students may also prepare special assignments
during the semester which may add a certain number of points to
their final grade.
Course objective. In this course students
should acquire fundamental knowledge of how the financial system and
its markets function. Basic participants and their influence on
interest rates and security prices will be considered. Students
should solve basic categories in financial systems. In seminars they
should analyse stock prices and interest rates and calculate yields
for these investments.
Prerequisites. Macroeconomics.
Course contents.
The economy and the financial system. The role of markets in the
economic system. Type of markets. Financial markets and the
financial system. Types of financial markets. The dynamics of the
financial system.
Financial assets, money and financial transactions. The creation of
financial assets. Lending and borrowing in the financial system.
Money as a financial asset. Types of financial transactions.
Interest rates in the financial system. Functions of the rate of
interest in the economy. The classical theory of interest rates. The
liquidity preference theory. The loanable fund theory.
Relationship between interest rates and security prices. Units of
measurement for interest rates and security prices. Measurement of
yield on a loan or security. Yield-price relationships.
Inflation, yield curve, duration and default risk influence on
interest rate.
The money market. Characteristics of the money market. The interest
rates in the money market. Money market securities.
Bond market. Principal features of corporate bonds. Basic
characteristics of corporate bonds. Yields on corporate bonds.
Stock market. Characteristics of stocks. Stock exchanges.
Teaching methods and student assessment.
Students are obliged to attend lectures and seminars. They will be
continuously assessed during the semester through tasks and two
tests. The final examination consists of the written and the oral
part.
Course objective. In this course students are
informed about differential calculus and integral calculus of
functions of several variables and of vector functions. Situations
in which a geometric view helps are primary analysed, i.e. real
functions of two or three variables, and the functions from R in R2
and R3. Lectures introduce and analyse basic notions, which are
illustrated by examples, while during exercises students adopt
corresponding techniques of approaching particular concrete problems
and solving them.
Prerequisites. Differential calculus, Integral calculus, Linear
algebra I.
Teaching methods and student assessment.
Lectures and exercises are obligatory for all students. During the
semester students can take tests that can replace the written
examination. The final examination consists of both a written and an
oral part and can be taken after the completion of lectures and
exercises.
Course objective. The objective of the course
at the introductory level based on geometry of plane and space is to
make students familiar with fundamentals of linear algebra.
Prerequisites. None.
Course contents.
1. Vectors in plane and space. Operations with vectors. Linear
dependence and independence of vectors. Basis of vector spaces.
Coordinate system. Norm of vectors. Distance between two points.
Cauchy - Schwarz - Buniakowsky inequality. Vector dot (scalar)
product. Direction cosine. Projection of vector to the straight line
and plane. Gramm - Schmidt orthogonalization process.
2. Square matrix of the second and third order. Square matrix of the
second and third order and their determinants. Orientation right
and left basis and coordinate systems. Vector cross product.
Algebraic properties of the vector product. Geometrical properties
of the cross product. Multiple vector-vector product. Jacobi
identity. Straight line and plane in space.
3. Linear operators in plane. Examples of operators: axial symmetry,
central symmetry, homothety, orthogonal projection, rotation. Basic
properties of the linear operator. Operations with linear operators
vector space L(X(M)). Products and power of the linear operator.
Matrix of the linear operator. Algebra of the matrix of the second
order. Contraction and dilatation of the plane eigenvectors and
eigenvalues of the linear operator. Symmetric linear operator in the
plane. Orthogonal linear operator in plane. Diagonalization of the
symmetric linear operator. Quadratic forms. Curves of the second
order.
4. Linear operators in space X_0(E). Transfer of all definitions
from plane. Existence of eigenvectors and eigenvalues. Orthogonal
linear operator. Symmetric linear operator. Surfaces of the second
order.
Teaching methods and student assessment. Lectures and exercises are
obligatory. The final examination consists of both a written and an
oral part that can be taken after the completion of all lectures and
exercises. During the semester students are encouraged to take 4 or
more tests that replace the written examination.
Course objective. Make students familiar with
nonverbal and verbal forms of communication. Train students towards
better oral and written communication in the forms necessary for
everyday and professional activities.
Teaching methods and student assessment.
Fundamentals of communications are based upon interactive learning
and interpersonal communication which is realised both in lectures
and in exercises. The final examination consists of both a written
and an oral part that can be taken after the completion of all
lectures and exercises. The final grade is influenced by the
following two segments: seminar paper and other students'
communication activities. To the final oral examination students
bring portfolios with all contributions written in that year.
Course objective. The goal of this course is to
introduce students to microcontroller (microprocessor) programming,
to make them understand working principles, embedding and
programming of computers in process control, as well as to practice
C-a and PIC programming. Lectures focus on working principles and
microcomputer programming and their connection in distributed
systems. During exercises students deal with techniques referring to
programming and design of computers in process control.
Prerequisites. Introduction to computer science, Programming
languages.
Teaching methods and student assessment.
Lectures and exercises are obligatory. Student's knowledge is
continuously assessed during the semester by means of tests and
homework. The final examination consisting of a written and an oral
part takes places upon completion of lectures and exercises.
Course objective. Make students familiar with
basic ideas and methods of graph theory. Basic components will be
taught and some of their applications will be given in lectures. In
exercises students will master techniques and methods of solving
tasks and apply them to real problems. The course is the same for
all branches.
Course objective. At the introductory level
students should be introduced to fundamental ideas and methods of
mathematical analysis, which represent the basis for many other
courses. During lectures basic terminology will be explained in an
informal way, their utility and applications will be illustrated.
During exercises students should master an adequate technique and
become trained for solving concrete problems.
Prerequisites. Differential calculus Students will become
familiar with basic principles and models of the financial law and
financial science. State revenues and incomes will also be analysed,
as well as instruments of financing. Since the Department of
Mathematics envisaged the branch Financial and business mathematics
as giving students knowledge of this field from various aspects, the
objective of this course is to supplement the study programme by
providing fundamental knowledge in the field of law and financial
science.
2. State revenues. Generally about incomes. Kinds of
state revenues. Fiscal incomes. Legal basis for the acquisition of
income. Taxes. The concept and characteristics of taxes. The tax
terminology – elements of taxation. Taxpayer. Base tax. Tax rate.
Border of taxation. Justification of levying taxes. Aims of
taxation. Effects and functioning of taxes. Tax evasion, tax
shifting and double taxation. Tax division. Principles of taxation.
The tax system. Taxation system in the Republic of Croatia. Taxation
of income, Taxation of profit. Contemporary economic theory and
practice of income taxation. Financing of the units of local
self-government. Purchase tax. Value added tax. The single purchase
tax, excise duties, excises. Rights and real property transfer tax
and. Tax on inheritances and gifts. Surveillance and controls. The
new system of state administration. Financial police. Government
audit. Commercial revision. Other public revenues. Customs - the
customs system of the Republic of Croatia. Contributions. Fees.
Concept of fees in the Republic of Croatia. Parafiscal incomes.
Nonfiscal incomes. Financing local and regional self-government.
3. Public loan. The concept of public loan and its
characteristics. Kinds of public loans. Conversion and reprogramming
of public loans.
4. Public expenditures. Concept and characteristics
of public expenditures. Kinds of state expenditures. Principles of
public expenditures. Causes of the growth of public expenditures.
Structure of public expenditures.
6. Monetary law. Types of banks. Banking operations.
Bank system of the Republic of Croatia. Loss of monetary
souvereignity. International monetary institutions.
Teaching methods and student assessment. All
students are obliged to attend lectures and exercises. Students are
occasionally assigned homework which influences their final grade.
They should also prepare a seminar paper and present it to other
fellow students. The final examination that can be taken at the
semester end consists of a written and an oral part.
Course objective. Introduce students to basic
applications of methods in simple mathematical models. Students will
be able to use basic results of mathematical physics, which will
allow them to apply numerical methods for calculation in some simple
mathematical models. Dynamical systems with one and more than one
degree of freedom will be considered Student's knowledge is continuously assessed during the
semester by means of tests and homework. The final examination
consisting of a written and an oral part takes places upon
completion of lectures and exercises.
Course objective. Make students familiar with
basic structures and methods of combinatorial and dicrete
mathematics. Basic components will be taught and some of their
applications will be given in lectures. In exercises students will
master techniques and methods of solving tasks and apply them to
real problems. The course is the same for all branches.
Course objective. Today necessary, classical
and relatively simple theory of functions of complex variable can be
presented in a such a way that a student adopts it as a completed
unity as well as a tool for solving a series of problems in
applications.
Teaching methods and student assessment.
Complex analysis related exercises should be mostly auditory. Strict
proofs will be mainly avoided. Lectures and exercises are obligatory.
Student's knowledge is continuously assessed during the semester by
means of tests and homework. The final examination consisting of a
written and an oral part takes places upon completion of lectures
and exercises.
Course objective. To develop techniques for
discrete objects in analogy with techniques for continuous objects.
Concrete mathematics is a blending of CONtinuous and disCRETE
mathematics. More precisely, it is a controlled manipulation of
mathematical formulas using a collection of techniques for solving
problems. The course is important in computer science for the
analysis of algorithms, since it deals with a collection of
fundamental mathematical facts. (A quote by I.M.Gelfand says:
"Theories appear and vanish but examples remain").
Prerequisites. A certain level of
mathematical maturity is necessary, no knowledge of combinatorics is
required, since everything is developed from the first principle.
Teaching methods and student assessment.
Lectures are based on selected topics from the [1] (containing a
number of additional topics accesible to those students who wish to
broaden their knowledge). Exercises will be carefully selected from
[1] (and given to students in advance). Book [1] includes more than
500 exercises, divided into six categories (complete answers are
provided for all exercises, except for research problems, making the
book particularly valuable for self-study). From time to time
summation techiques will be illustrated by using packages MAPLE and
MATHEMATICA. The examination will be written and oral. Student's
knowledge is continuously assessed during the semester by means of
tests and homework. Tests can replace the written part of the final
examination. Students are also encouraged to prepare seminar papers
which influence the final grade.
Course objective. Most important topics of
Euclidean geometry from the point of constructive methods are
treated in this course with necessary theoretical preliminaries.
Most of these subjects are covered from the analytical and
synthetical point of view in courses Analytic geometry and
Elementary geometry. Special stress in this course is put on the
application of constructive methods in geometric parts of teaching
in primary and secondary schools. During exercises students use
computer software with geometry contents.
Course contents.
1. Euclidean constructions. Constructive task. Methods of solving.
Algebraic method. Method of intersection. Method of transformation.
2. Isometries of Euclidean plane. Axes and central symmetries.
Translations and rotations. Glide symmetries. The group of
isometries and some of its subgroups.
3. Homothety and similarity. Power of the point with respect to the
circle. Potential axis and potential center. Inversion.
4. Projective mappings of the Euclidean plane. Double ratio.
Perspective collineations. Perspective affinity.
5. Conic sections. Ellipse, parabola and hyperbola. The plane
intersection of the cone and cylinder. Pascal and Brianchon theorem.
Conic sections as the perspective images of the circle. Ellipse as
the perspective affine image of the circle.
6. Constructions by means of limited instruments. Constructions by
means of the ruler. Constructions in a limited part of a plane.
Constructions by means of the ruler with the given auxiliary figure.
Steiner constructions. Construction by means of two-side ruler.
Hilbert-Bachman constructions. Mohr-Mascheroni constructions.
7. Non-elementary constructions. Constructibility by means of a
ruler and a compass. Duplication of the cube and angle trisection.
Non-elementary solutions of the duplication of the cube and angle
trisection. Quadrature of the circle. Approximative solutions of
three classical problems.
8. Elements of descriptive geometry.
Teaching methods and student assessment.
Lectures are auditory. Exercises are performed in groups in the
computer lab using computer software with contents in the field of
geometry. The final examination consists of both the written and the
oral part.
Course objective. Teach basic notions in
cryptography and protection of computer systems. Exercise projecting
of cryptography schemes. Make students familiar with multi-user and
multitask operating systems (UNIX, Linux, XP). Make students
familiar with notions from cryptography and protection of operating
systems. Students will learn various cryptography schemes and
implement them as well as analyse security of operating systems and
databases.
Teaching methods and student assessment.
Lectures and exercises are mandatory. Student's knowledge is
regularly assessed during the semester by means of homework and
tests. The final examination consisting of a written and an oral
part, is taken at the end of the semester.
Teaching methods and student assessment.
Attendance of classes and exercises is obligatory. Knowledge
assessment consists of three parts:
1. Points obtained through three preliminary tests held during the
semeste
2. Points obtained at the final examination
3. Oral part of the examination
Teaching methods and student assessment.
Attendance of classes and exercises is obligatory. Knowledge
assessment consists of three parts:
1. Points obtained through three preliminary tests held during the
semester
2. Points obtained at the final examination
3. Oral part of the examination
Course objective. To introduce students with
observation, modelling, solving and interpretation of real problems
of optimisation. To analyse the basic method for solving linear
programming problem – simplex method and to apply it as much as
possible to real problems from practice, using thereby a computer
and software Winqsb. Stress will be placed on observing a problem,
modelling and interpreting of results.
Teaching methods and student assessment.
Lectures and exercises are obligatory. The course is carried out in
the form of theoretical classes, by solving tasks and cases from
practice during exercises, as well as by working on computers and
using software package Winqsb. Students' knowledge is assessed by
means of homework, a written and an oral part of examination.
Course objective. The course aims to develop
understanding of the economy's behaviour over time, exploring the
causes of fundamental macroeconomic problems and evaluating the
effects of macroeconomic policy (primarily monetary and fiscal) on
the economy's behaviour and performance. This course encourages the
application of system and model approach in the analysis of economic
activities, problems and evaluation of macroeconomic performance and
policy.
Teaching methods and student assessment.
Class attendance is mandatory. Students are encouraged to actively
participate in class discussions and use Internet. Students'
knowledge is assessed continuously through in-class participation
and tests. The final examination is composed of an oral and a
written part.
Course objective. The course is focused on
studying mathematical models used for description and research of
various phenomena in biology. These and/or similar models are used
in other fields such as medicine, psychology, ecology, etc At the introductory level
make students familiar with ideas and methods of mathematical logic.
Basic components will be taught and some of their applications will
be given in lectures. In exercises students will master techniques
and methods of solving tasks and apply them to real problems. The
course is the same for all branches.
Prerequisites. Calculus I and Calculus II.
Course contents.
Propositional logic: Introduction. Syntax. Semantics. Normal form.
Theorem of compactness. Tests of worthiness. Adjustment of
proposition. Consistency. Natural deduction. Some other axiomatics
of propositional logic.
First order logic: Introduction. Syntax. Interpretations and models.
Main test. Adjustment of theories of first order. Theorem of
completeness. Some examples of first order theory.
Teaching methods and student assessment.
Lectures and exercises are obligatory. The final examination
consisting of a written and an oral part takes places upon
completion of lectures and exercises.
Teaching methods and student assessment.
Lectures and exercises are mandatory. Students' knowledge is
assessed by means of tests. The final examination which consists of
a written and an oral part takes place at the end of the semester.
Course objective. The course objective is to
introduce students to the fundamental concepts and methods in modern
financial mathematics. Students will be introduced to theory of
martingales and stochastic differential equations. Attention will be
placed on the applied examples and the intuitive and informal
understanding of models and theory. In the practical part of the
course students are supposed to learn the necessary techniques and
solve concrete problems using computers.
Prerequisites. Probability. Statistics. Random processes.
Teaching methods and student assessment.
Teaching will be performed in a series of lectures, exercise classes
and practical classes in the computer lab. Grading will be based on
the final examination (oral or written), homework assignments during
the semester and/or project work.
Course objective. To introduce students to
some elementary mathematical aspects of electoral systems, such as
evaluation and forming of the electoral systems, models of electoral
systems, basic methods of assignments, etc.
Teaching methods and student assessment. One
part of classes refers to classical lecture-type classes with basic
notions, characteristics, mathematical aspects and problems of
electoral systems explained and seminar paper topics proposed. The
other part of classes encompasses presentations of student seminar
papers covering topics falling into the scope of mathematical
aspects of electoral systems. The final examination consists of a
written and an oral part, and a seminar paper, which, if successful,
affects the final grade and can replace the oral examination either
partially or completely.
Course objective. Through lectures and
seminar papers students will become familiar with some classical
mathematical models described by ordinary differential equations
used in various fields of human activity (physics, engineering,
economy, medicine, biology, agriculture The objective of the course
is to introduce students to methodology of scientific research.
After having attended the course and passed the examination students
should be able to independently solve a given problem, research
literature, write and present their papers in an interesting way.
Prerequisites. Bachelor level degree in
mathematics.
Course contents.
Since every year several new topics are introduced, we will mention
several topics presented so far: Data generation and presentation.
Floating-point arithmetic. Interpolation. Linear and cubic least
squares spline. Best L_p (p \geq 1) approximation. L_p (p \geq 1)
distance between a point and a straight line and between a point and
a curve. Best least squares and best total least squares straight
line. Solving special systems of linear equations. LU decomposition
of a three-diagonal matrix. Iterative methods for solving large
systems of linear equations. Eigenvalue problem. Power method.
Solving the equation f(x)=0. Method for solving the system of
nonlinear equations. Horner's algorithm. Gauss-Newton method. One-dimensional
minimisation. Multiple dimensional minimisation. Golden section
search, parabolic interpolation and Brent's method. Application of
Nelder-Mead algorithm. Numerical method for solving differential
equations. Magic squares. Teaching methods and student assessment.
During lectures students are introduced to scientific-research and
professional activities by means of small projects. Every topic is
dealt with by doing the following: motivation and mathematical
elaboration, derivation of basic formulas, laying foundations for a
design of a Mathematica or Matlab program, programming and program
testing. Exercises are laboratory. They are done by using computers
and LCD projectors supported by Mathematica and Matlab software.
During classes students can take several tests. Every student is
also individually assigned a seminar paper. Students should first
study the topic by using some fundamental literature, then search
additional literature (books, articles from journals and on the
Internet) after which they explain the topic in front of their
teachers and fellow students. After that the topic and the way of
preparation are defined in detail. Seminar papers should be written
in \LaTeX, with an abstract both in Croatian and English, key words
in Croatian and English and AMS Mathematical Classification (2000).
The text should be divided into sections and subsections, and the
cited formulas should be marked appropriately. Literature, which is
mentioned at the end of the paper, should be listed in accordance
with AMS regulations and cited at least once in the text. Finally,
students are obliged to present their papers to teachers and other
students. A successful seminar paper replaces the written part of
the examination, and during seminar paper presentation in front of
all other students, the oral examination takes place. Successful
presentation of high quality guarantees a high grade. The paper will
be put on the Department's web page and enter into competition for
one of the awards (e.g. Rector's Award).
Course objective. The objective of this
course is to make students familiar with the main methods of one-dimensional
and multidimensional minimisation with or without constraints.
Minimisation methods of nondifferentiable functions will be analysed
in particular. Thereby proving theorems will be avoided, except in
case of some constructive proofs which themselves refer to the
construction of ideas or methods.
Prerequisites. Bachelor level degree in
mathematics.
Course contents.
1. Introduction. Local and global minimum. Illustrated examples from
applications. Convex functions.
2. One-dimensional minimisation. Golden section search, parabolic
interpolation and Brent's method. Newton's method and its
modifications.
3. Multidimensional minimisation without constraints. Gradient
method. Steepest descent method. Newton's method and its
modifications. Quasi-Newton methods. Conjugate gradient method.
Least squares problems. Examples and applications. Graphical
interpretations of an iterative procedure.
4. Multidimensional minimisation without constraints of
nondifferentiable functions (Searching methods). Method of
coordinate relaxation. Nelder-Mead downhill simplex method. Powell's
method. Methods of random search.
5. Nonlinear programming. Motivation and examples. Basic methods.
Teaching methods and student assessment. Lectures will be
illustrated by ready-made software packages and graphics using a PC
and an LCD projector by means of Mathematica or Matlab. Exercises
are partially auditory and partially laboratory, and students will
use PCs and an LCD projector by means of the aforementioned software
systems. The final assessment consists of both the written and the
oral examination that can be taken after the completion of all
lectures and exercises. During the semester students are given
homework. They can also take 2-4 tests that completely cover course
contents. Successful tests replace the written examination. Students
are encouraged to prepare seminar papers. Successful seminar papers
influence the course final grade.
Course objective. To make students familiar
with current approaches to lecture structure and organisation,
teaching aids, and procedures used in teaching computer science. To
teach students how to acquire knowledge on the basis of which they
will be able to keep up with a rapid and pervasive development of
computer science and communications as well as to introduce new
procedures and aids in teaching computer science.
Teaching methods and student assessment.
Lectures and exercises are obligatory. Lectures are aimed at
pointing out to methodological and didactic principles applicable to
computer science teaching, and linking them to available tools and
software solutions. Laboratory exercises and seminar papers are
directed towards solving real problems in a (computer) classroom
based upon the mentioned principles. The final examination consists
of a written and an oral part. During the semester students can
choose one of the proposed topics for their seminar paper that
should be relative to the course contents. A successfully prepared
and presented seminar paper represents an equal part in defining the
final grade.
Course objective. To introduce methods of
teaching primary and secondary school mathematics and develop
teaching skills.
Prerequisites. Bachelor level degree in
mathematics.
Course contents.
Guiding principles in teaching mathematics (respect, preparation,
clarity, active learning).
Teaching forms (frontal, individual, work in groups, work in pairs,
tutorial work; individual and team projects, work on problems, work
with text and other media, etc. ).
Inductive vs. deductive method.
The role of mathematical theory. (How to introduce a new notion in
order to satisfy prescribed aims and objectives and pupils' age,
theorems and proofs in teaching mathematics)
Exercises. (Methodology of solving different types of excesses -
algebra, geometry, problem, etc.)
Homework.
Assessment in mathematics (State of pupils' knowledge at the
beginning, assessing pupils' progress – observations and checklists,
test construction, assessing a teacher)
Integrating technology in mathematics instructions. (Models,
overhead projector, videos, computer technology and internet)
Planning. (Planning a course, unit planning, lesson planning,
reflecting on a lesson)
Examples of teaching mathematics (Attend and analyse several lessons
at primary and secondary schools, discuss ideas about teaching
methods referring to specific mathematical topics)
Teaching methods and student assessment.
Students should attend lectures and prescriptive classroom
activities in primary and secondary school, do their homework and
seminar. The final examination is oral. The final grade is a
combination of grades obtained in seminars, homework and the final
examination.
Course objective. The objective of this
course is to educate and train students to be able to apply modern
and traditional didactic strategies and methods in teaching primary
school mathematics. The possibility of applying some strategies and
teaching methods in dependence of mathematical subjects necessary to
be learnt, will be studied, by combining lectures, practice and
individual projects depending on the age and ability of students, as
well as the aims of secondary schools in question. Special stress is
placed on working with pupils showing special interest in
mathematics, competitions in mathematics, working with pupils having
difficulties in mathematics and visualisation of mathematics.
Prerequisites. Teaching mathematics I.
Course contents.
Methodology and subjects of the work with gifted pupils.
Competitions in mathematics. Reporting from professional-methodic
journals and journals for the secondary school pupils. Preparation
for writing a professional paper the subject of which falls into the
scope of working with pupils showing special interest in mathematics.
Preparation for the presentation of a paper at professional
conferences the topic of which falls into the scope of working with
pupils showing special interest in mathematics. Creating project
tasks which can be used in project teaching. Inventing materials
which can be used in programmed teaching. Making posters,
presentations and some other materials for the purpose of
visualisation and popularisation of mathematics. Methodology of
special subjects in vocational secondary schools of e.g. commerce,
civil, mechanical engineering, etc. Analysis of subjects and other
materials for teaching mathematics in other countries.
Teaching methods and student assessment.
Lectures and exercises are obligatory. Exercises are performed in co-operation
with secondary schools. Students must attend, analyse and perform
the arranged lectures led by the course assistant co-operating with
the secondary shool maths teacher as a tutor. Student assessment is
carried out regularly. Mathematical classes students perform at
schools are also assessed. The final assessment consists of the oral
examination that can be taken after the completion of all lectures
and exercises.
Teaching methods and student assessment.
Students should regularly attend lectures and exercises. The final
assessment which consists of the written and the oral part can be
taken after the completion of lectures and exercises.
Course objective. This course covers the core
concepts and methods of microeconomic analysis, using some
mathematics in modeling and explication. By the end of the course
unit students should be able to understand and apply basic
microeconomic principles to the economic decisions of households and
firms under a variety of market conditions. The aim of this unit is
to enable students to deepen their analytical ability in
microeconomics so that they can use theory to generate predictions
and explanation with respect to economic phenomena.
Prerequisites. None.
Course contents.
1. Introduction. The concept and topics of microeconomic theory.
Microeconomic entities.
2. Market determinants of micro economy. Products and services
market. The concept and factors of demand. Elasticity of demand.
Supply in competition. The supply changes factors. Elasticity of
supply. Dynamics of competition. Labor market. Capital market.
3. Production functions. Allocation of economic resources.
Production possibilities curve. Short run production function.
Total, average and marginal return of the production resource. Law
of diminishing returns. Three stages of production function.
Combining production resources.
4. Costs of production. Opportunity and actual costs. Economic and
accounting concept of costs. Theory of costs in the short run. Fixed
costs. Variable costs. Total costs function. Average and marginal
costs. Short run costs curve. Costs in the long run.
5. Profit maximisation. Total, average and marginal revenue.
Production and revenue interaction. Profitability of resource
investment. Normal and economic (pure) profit. Total and average
profit. Marginal revenue and marginal profit. Maximal profit and
market prices. Costs and revenues in different market structures.
Break-even point. Concept and application of marginal analysis.
6. Theory of investment. Concept of investment. Static and dynamic
decision models. Demand of the firm for investments. Decisions under
subjective risks. Risk identification. Risk measurement. Valuation
of risky investment projects. Decision under uncertainty. Risk
analysis using simulation.
Teaching methods and student assessment.
Students are required to attend lectures and exercises on a regular
basis. During the course students will be continuously assessed by
means of homework and tests. The final examination which consists of
the written and the oral part can be taken after the completion of
lectures and exercises.
Course objective.
The objective of the course is to make students familiar with basic
principles and models of new product diffusion. Features of the
given models will be analysed and fundamental methods for parameter
estimation in models will be presented.
Teaching methods and
student assessment. Lectures
will be illustrated by ready-made software packages and graphics
using a PC and an LCD projector by means of Mathematica or Matlab.
The final assessment which consists of the written and the oral part
can be taken after the completion of lectures and exercises. During
seminars students will be assigned small projects for independent
work that have to presented either to other students or at some
professional or scientific conferences. The final examination can be
taken after the completion of lectures and presentation of the
seminar paper that has to be presented to students and teachers
prior to taking the examination.
Course objective. To introduce students to
basic concepts in multimedia programming and to help them understand
COM, .NET objects and ActiveX controls. To exercise programming (Visual
C, Flash) of multimedia content in visual environment. Lectures
present graphics, animation, text and sound. Data compression
algorithms as well as data streaming algorithms are taught. During
laboratory exercises students should master techniques of visual
programming.
Teaching methods and student assessment.
Lectures and exercises are obligatory. During the semester students'
knowledge is continuously assessed by means of tests and homework.
The final examination consisting of a written and an oral part takes
places upon completion of lectures and exercises.
Teaching methods and student assessment.
Students should attend lectures, do their homework and seminars. The
final examination is oral. The final grade is a combination of
grades obtained in seminars, homework and the final examination.
Course objective. Make students familiar with
the basic idea and methods of numerical linear algebra which can be
applied to solving linear systems, least squares problems,
eigenvalue and singular value problems. Further, through lectures
students will be introduced to the usage of dense and sparse
matrices, floating point arithmetics and different matrix
factorisations as well as to the corresponding algorithms for
solving different problems in applications. Through exercises
students will deal with techniques for solving concrete problems by
using ready-made software packages or their own programmes.
Teaching methods and student assessment.
Students are obliged to attend lectures and exercises. During
exercises students will become trained for solving problems from the
fields of biology, chemistry, physics and engineering by using ready-made
software packages or by making their own programmes. Student's
knowledge is continuously assessed during the semester by means of
tests and homework. The final examination consisting of a written
and an oral part takes places upon completion of lectures and
exercises.
Course objective. The objective of this
course is to make students familiar with the main methods of
numerical mathematics, whereby proving theorems will be avoided,
except in case of some constructive proofs which themselves refer to
the construction of ideas or methods.
Course objective. Students should acquire
fundamental terminology from the fields of mathematics and computer
science as well as apply structures typical of GSP (German for
Specific Purposes). They should be taught and trained how to read
various pieces of literature pertaining to mathematics and computer
science as well as to carry out conversation referring to some basic
topics GSP (German for Specific Purposes).
They should be taught and trained how to read and understand various
pieces of literature pertaining to mathematics and computer science,
discuss topics in their fields of study and translate simple GSP
texts from Croatian into German. Students should also be taught how
to individually present a selected topic in Germanindividually presents a selected GSP topic, which altogether affects
their final grade. Students' knowledge is continuously assessed by
four tests, two per each semester, and the oral part of the
examination takes place at the end of academic year.
Course objective. To make students familiar
with the concept and a geometrical sense of an ordinary differential
equation. To show basic types and methods for solving. To make
students familiar with the existence and uniqueness theorems by
giving motivation only, without any precise proof. To present the
concept and basic methods for solving partial differential equations.
To illustrate concepts and methods by using numerous geometric and
practical examples by means of a computer course consists of two
mutually independent units: Lebesque integral and Inequalities. The
goal of the course is to make students familiar with fundamental
concepts pertaining to both of these areas applicable in mathematics
and engineering Make students familiar with
pedagogy as a science of education. Enable students to develop their
creative thinking and improve their educational practice and
pedagogic theory.
Prerequisites. Not required.
Course contents.
Man, education, society. Pedagogy as a criticising and creating
science. System of pedagogic sciences.Theory and practice of
pedagogy. Development of pedagogy – general and national history of
pedagogy. Future of pedagogy.
Pedagogic methodology. Types of research. Scheme. Hypotheses and
variables. Instruments and procedures. Quantitative vs. qualitative
analysis. Research.
Teaching process analysis. Goal, tasks, ideals. Models of
concretising education objectives. Functional and international
education.
Education. Fields of education. Educational factors and their
influence. Education place and education-specific characteristics.
Principles, methods, procedures, instruments and forms of education.
School, management, education policy.
The meaning and history of schools. Theories. Education systems in
Croatia and abroad. Teachers and their competencies. The meaning and
importance of education policy. Management theories, models and
procedures.
Teaching methods and student assessment.
Students are obliged to attend lectures and exercises. Every student
is expected to prepare a seminar paper and conduct two exercises.
The final examination consists of both a written and an oral part
that can be taken after the completion of all lectures and
exercises.
Course objective. The objective of this
course is to make students familiar with fundamental operational
research methods and their applications. They would also be
introduced to related available software. Special stress will be
placed on problem observation, modelling and interpretation of
results.
Teaching methods and student assessment.
Lectures and exercises are obligatory. Lectures are carried out
theoretically, during exercises students solve problems and cases
from practice and use computers supported by the Winqsb software
package. Students' knowledge is assessed by means of assignments,
written and oral examination.
Course objective. To introduce students to
basic ideas and applications of Fourier analysis. Lectures are
presented by using examples from technology and physics. Results are
given with ideas of proofs by means of graphs, frequency diagrams
and geometric analogies.
Teaching methods and student assessment.
Lectures are based on examples. Exercises will be mainly laboratory
ones. Students' knowledge will be assessed continuously during
lectures. Test results make 50% of the final grade. The final
examination consists of the written and the oral part.
Course objective. Students will achieve
competences and skills in management, get an insight in new trends
in business and management, increase effectiveness of managing
career and personal professional growth, develop capabilities of
managing particular business situations, and increase personal
employability and personal competitive advantages on the labour
market. Management is presented as a practical skill as well as a
profession. The management role is often passed to the persons
without economic or managerial knowledge. Therefore students will
learn the basics of management in the business systems, basic skills
necessary for successful functioning in the organisational settings.
Some professions are highly focused on growing technical
competences, while social skills remain undeveloped. This course is
focused on development of social skills for future engineers who
will need to initiate, develop and maintain business contacts,
stimulative working conditions, personal and team motivation,
attraction and managing support groups for propositions and
solutions.
Teaching methods and student assessment.
Students will be evaluated during the semester through exercises,
simulations of business situations and case study analysis. The
final examination is due at the semester end. It is a written test
containing 30 questions (open type, closed type) followed by the
oral examination.
Course objective. To make students familiar
with the theory and application of artificial intelligence focusing
on prediction, classification and pattern recognition problems. The
methodology and architecture of neural networks, genetic algorithms,
intelligent agents, robotics and other artificial intelligence
techniques will be explained through lectures. Excercises will cover
examples and usage of software tools, thus enabling students to
acquire basic principles of design and evaluation of intelligent
systems.
Teaching methods and student assessment.
Lectures and excercises are obligatory. Student knowledge is
assessed during the semester through homework and project
assignments. The final examination consists of the written and the
oral part.
Course objective. Introduce students with the
models formed from partial differential equations which describe
natural phenomena. Further, introduce them with the basic techniques
for solving partial differential equations including separation of
variables and expansion using the eigenfunctions. The following will
also be considered: the method of characteristics, Fourier and
Laplace transformations and Green functionCourse objective. The course introduces
students to the development of major mathematical ideas in the
history. In that way they learn various examples useful for their
future careers as teachers of mathematics and for the communication
with people in other professions, they are introduced to connections
between mathematics and other professional fields as well as the
role of mathematics in the development of the human society, and
they revise mathematical facts they learned before. In the first
semester the development of mathematics in various cultures until
the renaissance is covered mostly chronologically, and in the second
semester mathematics in the period from the beginning of the 17th
until the beginning of the 20th century is covered by explaining the
development of major mathematical disciplines.
Prerequisites. Bachelor level degree in
mathematics.
Course contents.
First semester:
1. Early mathematics: Egyptian and Babylonian mathematics.
2. Mathematics in the Greek and Roman world: preeuclidean
mathematics (Thales, Pythagoreans, etc.), Euclidean age (Euclid,
Archimedes, etc.), posteuclidean mathematics (Ptolemy, Heron, etc.),
mathematics in the Roman state.
3. Mathematics of Eastern cultures: Indian and Chinese mathematics.
4. Medieval mathematics: Arabian mathematics, mathematics in
medieval Europe.
5. Renaissance: development of mathematical notation, solution of
algebraic equations of the 3rd and 4th degree, discovery of
logarithms, connections of mathematics and physics, astronomy and
arts.
Second semester:
1. Development of algebra after the renaissance: beginnings of group
theory, matrix theory, vector spaces; fundamental theorem of
algebra; development of number theory.
2. Development of analysis after the renaissance: discovery and
development of calculus; convergence, series, continuity; complex
numbers.
3. Development of geometry after the renaissance: discovery of
projective, analytical and non-euclidean geometries; beginnings of
topology.
4. Probability theory: beginnings and development until the
axiomatisation.
5. Creation of set theory.
Teaching methods and student assessment. The
attendance of classes is obligatory. During the course, every
student is required to prepare and present one seminar paper on a
given subject. The attendance at the seminar presentations is
obligatory. Plagiarism is strictly forbidden. Oral examination is
taken at the semester end.
Course objective. To make students familiar
with the process of modelling, designing, developing, implementing,
and maintaning educational software systems, especially web oriented
intelligent tutoring systems. Modern information technology enables
a more frequent usage of tutoring systems as an addition or a
substitute of the standard way of education. Those systems are
characterised with interactivity and adaptibility, including
hypermedia, Internet technologies, as well as artificial
intelligence. The course covers the process of designing intelligent
educational systems, as well as the usage of intelligent tutoring
system shells, artificial intelligence in tutoring systems, and
distance learning systems. Students will incorporate the process of
designing modules, such as the teacher module, the student module,
the communication module (i.e. user interface). Students will
therefore acquire the basic principles of design, development and
implementation of educational systems, as well as the skills of
evaluating such software.
Teaching methods and student assessment.
Lectures and excercises are obligatory. Lectures will cover theories
and models of learning, as well as technologies used for designing
and developing educational computer models covering different
topics. Exercises will cover the usage of software tools to build
educational systems. Students' knowledge is assessed during the
semester through homework and project assignments. The final
examination consists of a written and an oral part.
Course objective. Students will become
familiar with various aspects of children's growth and development
as well as the structure and personality development. They will also
acquire knowledge of psychology that might influence understanding
of the education practice.
Course objective. Students should acquire
knowledge that will help them understand, identify and develop
gifted pupils.
Prerequisites. Psychology of education.
Course contents.
Giftedness and talent – definition, characteristics, fields.
Acceleration of gifted children (purpose, procedure). Support in
education for gifted and talented pupils, individualisation of
education, small group work. Creativity. Maturity and selection of
occupations for gifted pupils.
The role of the teacher in the development of giftedness at school.
Desirable qualities of teachers of gifted pupils. Creating a
positive and supporting classroom atmosphere. Setting up a creative
atmosphere. Teacher and class mates attitudes to gifted pupils.
Teaching methods and student assessment.
Students are obliged to attend lectures and seminars. Every student
is expected to prepare a seminar paper. The final examination
consists of both a written and an oral part that can be taken after
the completion of all lectures and exercises.
Course objective. The purpose of this course
is to teach students how to use highly sophisticated mathematical
software and how to make their own mathematical programmes with this
software (Matlab, WRI Mathematica and Mapple). Students will be
introduced to different problems from mathematics, physics, economy
and engineering. Through lectures they will learn about software
packages and how one can write his/her own programme. Through
exercises they will learn how to solve specified problems from
different applications (numerical, symbolic or graphics).Students are obliged to attend lectures and exercises. Students'
knowledge is continuously assessed through tests and homework. In
lectures students study principles of computer networks and their
services (ftp, web services, RPC, P2P). In exercises students should
become able to solve programming techniques (java servlets, parsers,
XSLT) and acquire some skills referring to usage of network services
and protocols. They should also use this knowledge to program mobile
equipment (WML/WAP) and PDA (personal digital assistant). The final
examination consists of both a written and an oral part that can be
taken after the completion of all lectures and exercises.
Course objective. The objective of this
course is to teach students how to solve complex and network
programming tasks through team work. This practicum is the link
between all previously attended computer-related courses. Throughout
lectures students will find out more information about relationships
between mathematical tools and the web environment. In exercises
they will learn how to solve two specific projects and how to
publish them on the web. Projects may cover methodical problems in
mathematics and physics, which enables their usage primary and
secondary school education An introduction into basic
ideas and examples of stochastic processes at the level of a first
course in processes. Attention is focused on models of processes in
various branches of science. Lectures are to be given in an informal
way, illustrating their utility and applications. Exercises should
enable students to become able to master different techniques and
solve particular problems.
Teaching methods and student assessment.
Lectures and seminars are obligatory. During the semester students
are encouraged to take tests. The final examination consists of both
a written and an oral part.
Course objective. To unify contents of
computer science courses attended during studies, especially the
ones referring to programming. To study systematically methods and
tools for software development, ways of managing software
development projects as well as evaluation of costs.
Teaching methods and student assessment.
Lectures and exercises are obligatory. Lectures elaborate in detail
methods and tools for software development as well as models of
evaluation of costs. During exercises students work in project
teams. On the basis of theoretical knowledge and programming skills
acquired in previous courses, students carry out simpler independent
tasks, which make a project as a whole. Students' activity is
continuously assessed and evaluated, and the level of their overall
knowledge is tested and graded by the final oral examination.
Teaching methods and student assessment.
Students should attend lectures, do their homework and seminars. The
final examination consists of the written and the oral part. The
final grade is a combination of grades obtained in seminars,
homework and the final examination.
Course objective. Teach student various kinds
of simple and complex data structures and algorithms. Apply these
structures and algorithms by means of the object-oriented
programming language. Show the influence of a data structure on the
design and efficiency of algorithms and computer programmes.
Teaching methods and student assessment.
Students should attend lectures and exercises. Students learn
important simple and complex data structures (lists, trees, graphs,
etc.) and related algorithms (trees and graphs traversals, sorting
and searching, etc.). During exercises students solve algorithm
techniques by using the object-oriented programming language.
Student's knowledge is continuously assessed during the semester by
means of tests and homework. The final examination consisting of a
written and an oral part takes places upon completion of lectures
and exercises.
Course objective. Make students familiar with basic
ideas and methods of game theory.
Prerequisites. Bachelor level degree in mathematics.
Course contents.
1. Introduction. Basic concepts, motivation, definitions.
2. Strategies. Some special game theory strategies are considered as
well as optimal answers to each of them. Maxmin strategy and Minmax
strategy are pointed out. Various games are studied, spacially game
solutions with the sum equal to zero, 2xn and mx2 games. Domination.
Symmetric games and games similar to the game of poker. Minimum and
maximum strategy and games with the sum which differs from zero.
Mixed strategies of nonzero-sum games. Mixed Nash equilibria in 2x2
nonzero-sum games.
Teaching methods and student assessment.
Students
should obligatorily attend lectures and exercises. Basic concepts
will be considered and their applications illustrated. In exercises
students will master techniques and methods of solving tasks and
apply them to real problems. The final examination consisting of a
written and an oral part takes places upon completion of lectures
and exercises.
Purpose of the course. Make students familiar
with basic ideas and methods in decision theory. Students will be
motivated for studying standard problems of decision theory through
illustrative examples. They will learn basic methods for solving
standard problems of decision theory.
Teaching methods and student assessment.
Lectures will be illustrated by using ready-made software and an LCD
projector. Students will be assigned practical seminar papers.
Student's knowledge is continuously assessed during the semester by
means of tests and homework. The final examination consisting of a
written and an oral part takes places upon completion of lectures
and exercises. Seminar papers may replace the written part of the
final examination.
Content:
For most differential equations it is unlikely that exact solutions
can be found. However, numerical methods can give excellent
approximations. This course introduces the basic ideas and shows how
can be applied for particular cases. It covers the generation and
propagation of roundoff errors, convergence criteria and efficiency
of computations. Includes these topics:
1. Introduction. Examples of approximation. First-order difference
operators. Second-order difference operator and its properties. The
elimination method. A sample of a difference scheme.
2. Basic concepts. Introducing the difference scheme. Approximation.
Correctness. Convergence. The relation between those concepts. A lot
of examples.
3. Difference schemes for the heat-conduction equation. The forward
scheme. The implicit scheme. Schemes with variable weight factors.
Three-layer difference schemes.
4. Stability theory. Classes of difference schemes. Stability of
two-layer difference schemes in various spaces. Reducing three-layer
difference schemes to the two-layer ones. Stability of three-layer
difference schemes. Schemes with non-autoadjoint operators.
5. Difference schemes for the transfer equation. Two-layer schemes
with weight factors. Schemes with a conditional approximation.
6. Difference schemes for the vibrations of a string. Two-layer
schemes with weight factors. Methods for the improvement of the
approximation`s order.
7. Symmetrizable difference schemes. Classes of symmetrizable
difference schemes. Criteria for stability.
8. Asymptotic stability. Criteria for the asymptotic stability of
symmetrizable difference schemes. Example of stable, but not
asymptotic stable schemes.
Expected work: Throught the semester several
projects will be assigned, some involving programming skills. The
projects are normally undertaken by individuals. Each project is
followed by a class presentation.
Course objective. The objective of this
course is to prepare students – future teachers for working with
advanced pupils in mathematics, as well as for working referring to
preparing pupils for mathematical competitions. Through lectures and
seminar papers, various fields of mathematics will be included with
contents suitable for primary and secondary schools.
Teaching methods and student assessment.
Students should attend lectures and exercises. They should present
their seminar papers in which they analyse certain mathematical
topics, and at the same time they choose examples and tasks intended
for appropriate age of primary and secondary school pupils. Students
obtain the final grade on the basis of their written seminar paper
and its oral presentation, which also encompasses the oral
examination.
Course objective. The objective of the course
is to make students familiar with basic concepts, tools and models
in credit and credit risk analysing. The course is divided into
three parts: (I) Introduction – including definition of basic
concepts regarding credits and credit risks; (II) Credit risk models
– including the overview of certain credit risk models used in
financial institutions for making better business decisions, and
different input variables and methods used in establishing these
models. In this part students can also find an overiew of methods
used for developing models without mathematical arguments and
deductions, including the explanation of the purpose of every
method; (III) Implementation – including an overview of basic steps
that are necessary to make in succesful model implementation.
Teaching methods and student assessment.
Teaching methods used during the course are lectures and seminars,
with a combination of group discussions and case analyses.
Attendance and activity at both lectures and seminars are required.
Students' knowledge will be assessed on a regular basis through
tests and various assignments. Furthermore, students have to pass
the final examination which will be in the written and the oral
form. Students are encouraged to work on the project which will
represent the application of defining models on the example, and
influence the final grade.
Course objective. Develop general and
specific knowledge and skills of students referring to office
operations in modern business conditions. Apart from the general
framework of office operations, their development, structure and
processes, students should learn about the standards and norms of
office operations, principles of modern business communication, and
acquire skills required for practical application of digital
technology in business operation systems.
Teaching methods and student assessment.
Exercises and seminars are obligatory. Students' knowledge is
assessed during the year through assignments in which students need
to solve practical problems in office operations. Each assignment is
designed to cover one segment of office operations (from data
processing by text processor Word, table calculator Excel, to
programmes for presentations, communication, and time management MS
Outlook etc.). The final grade is given on the basis of the average
assignment grade and the oral examination.
Course objective. The objective of this
course is to make students familiar with the main ideas and
mathematical models of the money, bond and stock market. Where
possible, the theory will be illustrated by practical examples from
banking practice and other fields of application.
Teaching methods and student assessment.
Lectures will be illustrated by ready-made software packages and
graphics using a PC and an LCD projector by means of Mathematica or
Matlab. Exercises are partially auditory and partially laboratory.
Students will be expected to participate in lectures by doing
independent seminar papers. The final assessment consists of both
the written and the oral examination that can be taken after the
completion of all lectures and exercises. During the course students
can take 2-4 tests that completely cover course contents.
Successfully passed tests replace the written examination. During
their studies students are encouraged to prepare a seminar paper.
Successful seminar papers influence the course final grade.
Course objective. Teach students essentials
of computer programming in a procedural programming language. Train
them in producing simple algorithms and making programs based on
them in a procedural programming language. Develop in students the
way of thinking which will enable them to solve more complicated
algorithmic and programming problems. Teach students in solving
simple problems of numeric mathematical methods using self made
programs.
Teach students introductory knowledge in object oriented and visual
oriented programming. Give students prerequisites which will be
useful in further self education on computers.
Course objective. Make students familiar with
basic ideas and methods of computer science which will be the basis
for other computer courses. Simple examples of programmes in local
and web environment will be interesting for students and encouraging
in their future usage of computers. Throughout lectures students
will acquire knowledge of the basic concepts and their application.
During exercises students will master basic techniques of
programming (C and Java) on simple examples from standard and web
programming lectures
and exercises.
Course objective. The objective of this
course is to make students familiar with the main ideas and methods
of approximation theory. Special attention will be paid to
Chebyshev's approximations and least squares approximations. Where
possible, the theory will be illustrated by practical examples from
various fields of application. Students are expected to participate
in the teaching process by doing independent seminar papers objective of the course
is to give a more general view to the notions and results students
encounter in linear algebra courses during the first two years of
study. By a more abstract approach one should understand more deeply
and more clearly the matter basic for many modern mathematical
disciplines.
Teaching methods and student assessment.
Attending lectures and exercises is compulsory. The examination
consists of the written and the oral part which take place upon
completion of lectures and exercises. During the semester students
can take tests which replace the written part of the examination.
Teaching methods and student assessment.
Students should attend lectures, do their homework and seminars. The
final grade is a combination of grades obtained in seminars,
homework and the final examination.
Course objective. To introduce standard
notions of probability theory in order to understand and apply them,
especially through other courses based on probability (statistics,
random processes, time series analysis, multivariate analysis,
etc.).
Teaching methods and student assessment.
Students should attend lectures, do their homework and seminars. The
final grade is a combination of grades obtained in seminars,
homework and the final examination.
Teaching methods and student assessment.
Students should attend lectures and exercises. Students learn
important facts of statical and dynamical contents on the Web and
programming of client-server communication. Through exercises
students learn how to master Web programming by means of examples
from Javascript, Java applets, PHP/MySQL solutions and Web server
programming tools. Student's knowledge is continuously assessed
during the semester by means of tests and homework. The final
examination consisting of a written and an oral part takes places
upon completion of lectures and exercises.
Course objective. Teach students how to use
computers in science with applications in numerical analysis (linear
and nonlinear equations, integration, interpolation, simulations and
optimisations). Design, build programmes (C++ or Java) and analyse
parallel and sequential algorithms with good numerical properties.
Teaching methods and student assessment.
Students are obliged to attend lectures and exercises. In lectures
students study concepts of finite precision arithmetic, linear
algebra systems, PDE approximation, graphics design and elementary
methods of Hilbert space. In exercises students should design and
build programmes sequential and parallel programmes in those fields.
Students' knowledge is continuously assessed through tests and
homework. The final examination consists of both a written and an
oral part that can be taken after the completion of all lectures and
exercises.
Course objective. Professional Colloquium is held at
the Department of Mathematics, University of Osijek. It is primarily
intented for teachers of mathematics and computer science employed
at primary and secondary schools, as well as for students and all
others having interest in it. Professional colloquium initiates
communication between teachers of mathematics and encourages
experience exchange and contact with the students of final years.
The idea for Professional Colloquium developed from the Scientific
Mathematical Colloquium that has been continuously organized since
1993. The colloquium also hosts different professional workshops and
lectures related to mathematics and computer science as well as to
other fields in some way related to mathematics.
Teaching methods
and student assessment. Student is obliged to actively participate
in the activites of the Professional Colloquium of the Osijek
Mathematical Society. Regular seminar attending is confirmed with
lecturer's signature- moderator of the Professional Colloquium.
Professional Colloquium I Z009
(0+0+2) - 3 ECTS credits
Course objective. To develop students'
presentational skills. Improve the level of communication between
students and teachers of mathematics and computer science in primary
and secondary schools in the region. Course Contents. Professional
Colloquium is taught at the Department of Mathematics, University of
Osijek. It is primarily intented for teachers of mathematics and
computer science employed at primary and secondary schools, as well
as for students and all others having interest in it. Professional
colloquium initiates communication between teachers of mathematics
and encourages experience exchange and contact with the students of
mathematics. The colloquium also hosts different professional
workshops and lectures related to mathematics and computer science
as well as to other fields in some way related to mathematics.
Teaching methods and student assessment. Student is obliged to
actively participate in the activites of the Professional Colloquium
of the Osijek Mathematical Society. Regular seminar attending is
confirmed with lecturer's signature- moderator of the Professional
Colloquium.
Introduction to integration theory M058 (2+2+0)
– 4 ECTS credits
Course objective. The purspose of this course is to
get a deeper and clearer insight into integration theory which is
the basics for understanding many modern mathematical disciplines-3 tests that replace the written examination.
Teaching methods and
student assessment. Exercises are partially auditory, and partially
done in a lab using Mathematica programme system. The exam consists
of a written and oral part. During lectures students take 3-4
preliminary exams. Successfully completed preliminary exams can
replace the written of the exam. Students are also given homework,
and they can do a seminar, which influences the final grade.
Course objective. The purpose of this course is to
make students familiar with basic applications of stability and
efficacy of some standard algorithms of numerical linear algebra in
solving mathematical problems that appear in design and analysis of
linear dynamic systems as well as their control.
Teaching methods and student assessment. Lectures,
exercises and seminars are obligatory. The exam consists of written
and oral part and it is taken after completion of lectures and
exercises. Students can take 3 preliminary exams during the semester
which, if done successfully, replace the written part of the final
exam.
Course objective. To work on the basics of
reliability theory in mathematically formal way, but also with
emphasis on application
Prerequisites. Basics of statistics,
differential and integral calculus Note: It is expected that
students of Electrical Engineering and Mechanical Engineering
Faculty as well as Faculty of Food Technology will have interest in
this course.
Teaching methods
and student assessment. Lectures are obligatory. Lectures provide
basic concepts, mathematical aspects and problems in reliability
theory. The second part of lectures is used for giving student
seminars. Attendance at seminars is obligatory. The final exam
consists of written and oral part, and it is taken after the
completion of lectures. Students can hand in a seminar paper during
a semester. A successful seminar paper influences final grade and
can replace the oral part of the final exam. |
Summary: David Cohen's PRECALCULUS: A PROBLEMS-ORIENTED APPROACH, Sixth Edition, focuses on teaching mathematics by using a graphical perspective throughout to provide a visual understanding of college algebra and trigonometry. The author is known for his clear writing style and the numerous quality exercises and applications he includes in his respected books. In this new edition, graphs, visualization of data, and functions are now introduced much earlier and receive greate...show morer emphasis. Many sections now contain more examples and exercises involving applications and real-life data. While this edition takes the existence of the graphing calculator for granted, the material is arranged so that one can teach the course with as much or as little graphing utility work as he/she wishes. This Enhanced Edition includes instant access to Enhanced WebAssign the most widely-used and reliable homework system. Enhanced WebAssign users learn the basics of WebAssign quickly |
Combo: College Algebra with Student Solutions Barnett/Ziegler/Byleen/Sobecki College Algebra series is designed to give students a solid grounding in pre-calculus topics in a user-friendly manner. The series emphasizes computational skills, ideas, and problem solving rather than theory. Explore/Discuss boxes integrated throughout each text encourage students to think critically about mathematical concepts. All worked examples are followed by Matched Problems that reinforce the concepts being taught. New to these editions, Technology Connections illustrate how concepts that were previou... MOREsly explained in an algebraic context may also be solved using a graphing calculator. Students are always shown the underlying algebraic methods first so that they do not become calculator-dependent. In addition, each text in the series contains an abundance of exercises - including numerous calculator-based and reasoning and writing exercises - and a wide variety of real-world applications illustrating how math is useful. |
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that are both elegant and occasionally bizarre. This book introduces the reader to the concept of bignum algorithms and proceeds to build an entire library of functionality from the ground up. Through the use of theory, pseudo-code and actual fielded C source code the book explains each and every algorithm that goes into a modern bignum library. Excellent for the student as a learning tool and practitioner as a reference alike BigNum Math is for anyone with a background in computer science who has taken introductory level mathematic courses. The text is for students learning mathematics and cryptography as well as the practioner who needs a reference for any of the algorithms documented within.
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Meet the Author
Tom St Denis is the author of the industry standard LibTom series of projects. Tom is a senior software developer and cryptographer for the Advanced Micro Devices Corporation. He has been engaged in various international development contracts and speaking engagements since 2004. He is at work on his next book, Cryptography for Developers |
Matlab
A Practical Introduction to Programming and Problem Solving
By
Stormy Attaway, Boston University. PhD Boston University Department of Mechanical Engineering at Boston University, and Associate Chair for the Manufacturing Engineering undergraduate program within the department. She has been the course coordinator for the Engineering Computation courses at Boston University for over twenty years, and has taught a variety of programming courses using many different languages and software packages
Ancillaries available with the text:
Instructor solution manual (available Aug. 1st)
electronic images from the text (available Aug 16th)
m-files (available Aug 1st)
Audience Engineers learning to program and model in Matlab. Undergraduates in engineering and science taking a course that uses (or recommends) MATLAB.
Reviews
"In and updates to reflect current features and functions of the current release of MATLAB."--Lunar and Planetary Information "Assuming no knowledge of programming, this book presents both programming concepts and MATLAB's built-n functions, providing a perfect platform for exploiting its extensive capabilities for tackling engineering problems. The book starts with programming concepts such as variables, assignments, input/output, and selection statements, moves onto loops, and then solves problems using both the 'programming concepts' and the 'power of MATLAB' side by side. In, and updates to reflect current features and functions of the current release of MATLAB."--Lunar and Planetary Information Bulletin, December 2011, Issue 127, page 46 "This is the perfect book for anyone wanted to acquire a secure understanding of MATLAB fundaments and master its language. Many engineers and scientists now use MATLAB and Simulink to solve real-world problems. With the help of this book, they will be able to exploit the full power of MATLAB much sooner than they would using the online manuals, and be able to solve real problems much more quickly."--IEEE Electrical Insulation Magazine, page 70 |
This classroom activity presents College Algebra students with a ConcepTest and a Question of the Day activity concerning the...
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This classroom activity presents College Algebra students with a ConcepTest and a Question of the Day activity concerning the effect of the proportionality constant, k, on the y-intercept and position of an exponential graph where k>0 and C is an arbitrarily fixed value in f(x)=Ce^(kx
College Algebra or Liberal Arts math students are presented with two Questions of the Day and a write-pair-share activity...
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College Algebra or Liberal Arts math students are presented with two Questions of the Day and a write-pair-share activity involving Florida's population growth. The results show that students often do not have a clear grasp of the differences between linear growth and exponential growth.
College Algebra or Liberal Arts math students are presented with three Questions of the Day and a write-pair-share activity...
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College Algebra or Liberal Arts math students are presented with three Questions of the Day and a write-pair-share activity involving Florida's population growth (other states may be used in place of Florida). The results are quite revealing and show that while students may have learned how to perform the necessary calculations, their conceptual understanding concerning exponential growth may remain faulty. Student knowledge (or lack thereof) of the size of their state's population and its annual growth rate may also be surprising.
College Algebra or Liberal Arts math students are presented with a ConcepTest and a write-pair-share activity involving...
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College Algebra or Liberal Arts math students are presented with a ConcepTest and a write-pair-share activity involving Florida's population growth. The activity asks students to decide whether a ten-year growth rate can be divided by 10 to produce the corresponding annual growth rate for each of the ten years. The results show that, while students may have learned that exponential growth is a multiplicative process, their conceptual understanding concerning exponential growth is often a bit fuzzy.
College Algebra or Liberal Arts math students are presented with a Question of the Day and a write-pair-share activity...
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College Algebra or Liberal Arts math students are presented with a Question of the Day and a write-pair-share activity involving U.S. state population growth. Student knowledge (or lack thereof) of the annual growth rates of individual states may be surprisingThis problem-based learning activity helps students practice analyzing economic events associated with the following current...
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0130323136
9780130323132
Modern Geometries: Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically non-Euclidean in approach, it relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation. Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems. For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical scientists.
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Rent Modern Geometries 2nd edition today, or search our site for Michael textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson. |
Syllabus
Math 430, January 12 2009
Introduction Math 430 is an introduction to basic algebraic structures: groups, rings, and fields. Topics will include permutation groups, group and ring homomorphisms, field extensions, and connections with linear algebra.
Prerequisites The prerequisite is Math 429, or equivalent mathematical maturity and experience with linear algebra.
Instructor:
Russ Woodroofe
Cupples I 114
Office hours: Tues 1:00pm - 2:00pm Thurs 1:30pm - 2:30pm + by appt
russw at math,wustl,edu
Textbook Herstein, Topics in Algebra, 2nd ed. I will cover much of chapters 2, 3, and 5.
Grading Your grade will be based on 2 midterm exams and one final exam, together with weekly homeworks and an occasional quiz, in the proportions
Midterm exams
20% each
Final exam
30%
Homework and quizzes
30%
I have scheduled the midterms for Feb 18 and April 1 (in-class). The final is scheduled by the University for May 6th, 10:30am - 12:30pm.
If you are taking the course Pass/Fail, you need to do the equivalent of C- work to pass. |
Mathematics in Action : An Introduction to Algebraic, Graphical, and Numerical Problem Solving of a three-part series, An Introduction to Algebraic, Graphical, and Numerical Problem Solving, Fourth Edition, illustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities And The accompanying practice exercises. Along with the activities And The exercises within the text, MathXL® and MyMathLab® have been enhanced to create a better overall learning experience For The reader. Technology integrated throughout the text helps readers interpret real-life data algebraically, ... MOREnumerically, symbolically, and graphically. The active style of this book develops readers' mathematical literacy and builds a solid foundation for future study in mathematics and other disciplines: Twelve Keys to Success |
Discrete Mathematics Using a Computer
9781846282416
ISBN:
1846282411
Pub Date: 2006 Publisher: Springer
Summary: 'Discrete Mathematics Using A Computer' offers a new, 'hands-on' approach to teaching discrete mathematics. A simple functional language is used to allow students to experiment with mathematical notations which are traditionally difficult to pick up.
Odonnell, John is the author of Discrete Mathematics Using a Computer, published 2006 under ISBN 9781846282416 and 1846282411. Three hundred fifty one Discrete ...Mathematics Using a Computer textbooks are available for sale on ValoreBooks.com, eleven used from the cheapest price of $28.80, or buy new starting at $59.63 |
tential for simulations of systems a little too complicated for complete mathematical analysis, and thus is ideal for teaching simulation as a tool for understanding.
Many topics in biology interact with the engineering viewpoint in such a fashion.
Mathematics and Computer Science
RECOMMENDATION #1.5
Quantitative analysis, modeling, and prediction play increasingly significant day-to-day roles in today's biomedical research. To prepare for this sea change in activities, biology majors headed for research careers need to be educated in a more quantitative manner, and such quantitative education may require the development of new types of courses. The committee recommends that all biology majors master the concepts listed below. In addition, the committee recommends that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes. They should be adept at using computers to acquire and process data, carry out statistical characterization of the data and perform statistical tests, and graphically display data in a variety of representations. Furthermore, students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases.
The elucidation of the sequence of the human genome has opened new vistas and has highlighted the increasing importance of mathematics and computer science in biology. The intense interest in genetic, metabolic, and neural networks reflects the need of biologists to view and understand the coordinated activities of large numbers of components of the complex systems underlying life. Biology students should be prepared to carry out in silico (computer) experiments to complement in vitro and in vivo experiments. It is essential that biology undergraduates become quantitatively literate. The concepts of rate of change, modeling, equilibria and stability, structure of a system, interactions among components, data and measurement, visualizing, and algorithms are among those most important to the curriculum. Every student should acquire the ability to analyze issues arising in these contexts in some depth, using analytical methods (e.g., pencil and paper), appropriate computational tools, or both. The course of study would include aspects of probability, statistics, discrete models, linear algebra, calculus and differential equations, modeling, and programming. |
NCES to Examine Content Of Algebra 1 Courses
Long considered a crucial portal to the world of postsecondary
education and a launching point for more complex studies of
mathematics, Algebra 1 is at the heart of most students' academic
schedules in the late middle grades or early in high school.
Despite its increasing importance in schools, though, many observers
say there is little uniformity in how that course on the relations and
properties of quantities is taught, or what students are expected to
learn while taking it.
Troubled by that lack of knowledge, officials at the National Center
for Education Statistics are expected soon to launch a
first-of-its-kind study of introductory algebra, aimed at exploring the
content and... |
Obviously a good knowledge of maths is essential for good programming, my maths isn't the best ( I didn't pay attention in school all those years ago) so, in order to get one step closer to being the best I need some advice. What maths should I learn for specific areas of game programming?
If someone could outline some subjects and say one or two words about what areas of game programming they would be used in I'd greatly appreciate it. I'm working through the Khan academy programme at the moment, and I'm just getting into algebra now.
1 Answer
Learn the math as you need it. That's how I did, and it will "save you time". The math needed depends a lot on the game you are making (whether or not it's 2D or 3D, whether or not it has advanced physics, etc etc). |
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Description/Abstract
The acquiring of formal, abstract mathematical concepts by students may be said to be one of the major goals of mathematics teaching. How are such abstract concepts acquired? How does this formal knowledge interact with the students' intuitive knowledge of mathematics? How does the transition from informal mathematical knowledge to formal mathematical knowledge take place? This paper reports on a research project which is examining the nature of the interaction and possible conflict between the formal and the intuitive components of mathematical activity. Details are presented of an initial study in which mathematics graduates, who could be considered to have acquired formal mathematical concepts, tackled a series of geometrical problems. The study indicates the complex nature of the interaction between formal and intuitive concepts of mathematics. The plans for the next stage in the research project are outlined.
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The pagination of this final proof copy is exactly as it appears in the published version. |
CONSULTING EDITOR George Springer, Indiana University COPYRIGHT 0 1969 BY XEROX CORPORATION .All rights reserved. No part of the material covered by this copyright may be produced in any form, or by any means of reproduction. Previous edition copyright 0 1962 by Xerox Corporation. Librar of Congress Catalog Card Number: 67-14605 ISBN 0 471 00007 8 Printed in the United States of America. 1098765432
ToJane and Stephen
PREFACE This book is a continuation of the author's Calculus, Volume I, Second Edition. Thepresent volume has been written with the same underlying philosophy that prevailed in thefirst. Sound training in technique is combined with a strong theoretical development.Every effort has been made to convey the spirit of modern mathematics without undueemphasis on formalization. As in Volume I, historical remarks are included to give thestudent a sense of participation in the evolution of ideas. The second volume is divided into three parts, entitled Linear Analysis, NonlinearAna!ysis, and Special Topics. The last two chapters of Volume I have been repeated as thefirst two chapters of Volume II so that all the material on linear algebra will be completein one volume. Part 1 contains an introduction to linear algebra, including linear transformations,matrices, determinants, eigenvalues, and quadratic forms. Applications are given toanalysis, in particular to the study of linear differential equations. Systems of differentialequations are treated with the help of matrix calculus. Existence and uniqueness theoremsare proved by Picard's method of successive approximations, which is also cast in thelanguage of contraction operators. Part 2 discusses the calculus of functions of several variables. Differential calculus isunified and simplified with the aid of linear algebra. It includes chain rules for scalar andvector fields, and applications to partial differential equations and extremum problems.Integral calculus includes line integrals, multiple integrals, and surface integrals, withapplications to vector analysis. Here the treatment is along more or less classical lines anddoes not include a formal development of differential forms. The special topics treated in Part 3 are Probability and Numerical Analysis. The materialon probability is divided into two chapters, one dealing with finite or countably infinitesample spaces; the other with uncountable sample spaces, random variables, and dis-tribution functions. The use of the calculus is illustrated in the study of both one- andtwo-dimensional random variables. The last chapter contains an introduction to numerical analysis, the chief emphasisbeing on different kinds of polynomial approximation. Here again the ideas are unifiedby the notation and terminology of linear algebra. The book concludes with a treatment ofapproximate integration formulas, such as Simpson's rule, and a discussion of Euler'ssummation formula.
'I11 Preface There is ample material in this volume for a full year's course meeting three or four timesper week. It presupposes a knowledge of one-variable calculus as covered in most first-yearcalculus courses. The author has taught this material in a course with two lectures and tworecitation periods per week, allowing about ten weeks for each part and omitting thestarred sections. This second volume has been planned so that many chapters can be omitted for a varietyof shorter courses. For example, the last chapter of each part can be skipped withoutdisrupting the continuity of the presentation. Part 1 by itself provides material for a com-bined course in linear algebra and ordinary differential equations. The individual instructorcan choose topics to suit his needs and preferences by consulting the diagram on the nextpage which shows the logical interdependence of the chapters. Once again I acknowledge with pleasure the assistance of many friends and colleagues.In preparing the second edition I received valuable help from Professors Herbert S.Zuckerman of the University of Washington, and Basil Gordon of the University ofCalifornia, Los Angeles, each of whom suggested a number of improvements. Thanks arealso due to the staff of Blaisdell Publishing Company for their assistance and cooperation. As before, it gives me special pleasure to express my gratitude to my wife for the manyways in which she has contributed. In grateful acknowledgement I happily dedicate thisbook to her. T. M. A.Pasadena, CaliforniaSeptember 16, 1968
Contents xvii10.3 Other notations for line integrals 32410.4 Basic properties of line integrals 32610.5 Exercises 32810.6 The concept of work as a line integral 32810.7 Line integrals with respect to arc length 32910.8 Further applications of line integrals 33010.9 Exercises 33110.10 Open connected sets. Independence of the path 33210.11 The second fundamental theorem of calculus for line integrals 33310.12 Applications to mechanics 33510.13 Exercises 33610.14 The first fundamental theorem of calculus for line integrals 33110.15 Necessary and sufficient conditions for a vector field to be a gradient 33910.16 Necessary conditions for a vector field to be a gradient 34010.17 Special methods for constructing potential functions 34210.18 Exercises 34510.19 Applications to exact differential equations of first order 34610.20 Exercises 34910.21 Potential functions on convex sets 350 11. MULTIPLE INTEGRALS11 .l Introduction 35311.2 Partitions of rectangles. Step functions 35311.3 The double integral of a step function 35511.4 The definition of the double integral of a function defined and bounded on a rectangle 35711.5 Upper and lower double integrals 35711.6 Evaluation of a double integral by repeated one-dimensional integration 35811.7 Geometric interpretation of the double integral as a volume 35911.8 Worked examples 36011.9 Exercises 36211.10 Integrability of continuous functions 36311 .I 1 Integrability of bounded functions with discontinuities 36411.12 Double integrals extended over more general regions 36511.13 Applications to area and volume 36811.14 Worked examples 36911. I5 Exercises 37111.16 Further applications of double integrals 373 11.17 Two theorems of Pappus 376 11.18 Exercises 377
1 LINEAR SPACES1.1 Introduction Throughout mathematics we encounter many examples of mathematical objects thatcan be added to each other and multiplied by real numbers. First of all, the real numbersthemselves are such objects. Other examples are real-valued functions, the complexnumbers, infinite series, vectors in n-space, and vector-valued functions. In this chapter wediscuss a general mathematical concept, called a linear space, which includes all theseexamples and many others as special cases. Briefly, a linear space is a set of elements of any kind on which certain operations (calledaddition and multiplication by numbers) can be performed. In defining a linear space, wedo not specify the nature of the elements nor do we tell how the operations are to beperformed on them. Instead, we require that the operations have certain properties whichwe take as axioms for a linear space. We turn now to a detailed description of these axioms.1.2 The definition of a linear space Let V denote a nonempty set of objects, called elements. The set V is called a linearspace if it satisfies the following ten axioms which we list in three groups. Closure axioms AXIOM 1. CLOSURE U N DER ADDITION . For every pair of elements x and y in V therecorresponds a unique element in V called the sum of x and y, denoted by x + y . AXIOM 2. CLOSURE UNDER MULTIPLICATION BY REAL NUMBERS. For every x in V andevery real number a there corresponds an element in V called the product of a and x, denotedby ax. Axioms for addition AXIOM 3. COMMUTATIVE LAW. For all x and y in V, we have x + y = y + x. AXIOM 4. ASSOCIATIVELAW. Forallx,y,andzinV,wehave(x+y) + z =x +(y+z). ?
4 Linear spaces AXIOM 5. EXISTENCEOFZEROELEMENT. There is an element in V, denoted by 0, such that x+0=x forallxin V . AXIOM 6. EXISTENCEOFNEGATIVES. For every x in V, the element (- 1)x has the property x+(-1)x= 0 . Axioms for multiplication by numbers AXIOM 7. ASSOCIATIVE LAW. For every x in V and all real numbers a and b, we have a(bx) = (ab)x. AXIOM 8. DISTRIBUTIVE LAW FOR ADDITION IN V. For all x andy in V and all real a,we hare a(x + y) = ax + ay . AXIOM 9. DISTRIBUTIVE LAW FOR ADDITION OF NUMBERS. For all x in V and all reala and b, we have (a + b)x = ax + bx. AXIOM 10. EXISTENCE OF IDENTITY. For every x in V, we have lx = x. Linear spaces, as defined above, are sometimes called real linear spaces to emphasizethe fact that we are multiplying the elements of V by real numbers. If real number isreplaced by complex number in Axioms 2, 7, 8, and 9, the resulting structure is called acomplex linear space. Sometimes a linear space is referred to as a linear vector space orsimply a vector space; the numbers used as multipliers are also called scalars. A real linearspace has real numbers as scalars; a complex linear space has complex numbers as scalars.Although we shall deal primarily with examples of real linear spaces, all the theorems arevalid for complex linear spaces as well. When we use the term linear space without furtherdesignation, it is to be understood that the space can be real or complex.1.3 Examples of linear spaces If we specify the set V and tell how to add its elements and how to multiply them bynumbers, we get a concrete example of a linear space. The reader can easily verify thateach of the following examples satisfies all the axioms for a real linear space. EXAMPLE 1. Let V = R , the set of all real numbers, and let x + y and ax be ordinaryaddition and multiplication of real numbers. EXAMPLE 2. Let V = C, the set of all complex numbers, define x + y to be ordinaryaddition of complex numbers, and define ax to be multiplication of the complex number x
Examples of linear spacesby the real number a. Even though the elements of V are complex numbers, this is a reallinear space because the scalars are real. EXAMPLE 3. Let V' = V,, the vector space of all n-tuples of real numbers, with additionand multiplication by scalars defined in the usual way in terms of components. EXAMPLE 4. Let V be the set of all vectors in V, orthogonal to a given nonzero vectorIV. If n = 2, this linear space is a line through 0 with N as a normal vector. If n = 3,it is a plane through 0 with N as normal vector. The following examples are called function spaces. The elements of V are real-valuedfunctions, with addition of two functions f and g defined in the usual way: (f + g)(x) =f(x) + g(x) for every real x in the intersection of the domains off and g. Multiplication of a functionf by a real scalar a is defined as follows: af is that function whose value at each x in the domain off is af (x). The zero element is the function whose values are everywhere zero. The reader can easily verify that each of the following sets is a function space. EXAMPLE 5. The set of all functions defined on a given interval. EXAMPLE 6. The set of all polynomials. EXAMPLE 7. The set of all polynomials of degree 5 n, where n is fixed. (Whenever weconsider this set it is understood that the zero polynomial is also included.) The set ofall polynomials of degree equal to IZ is not a linear space because the closure axioms are notsatisfied. For example, the sum of two polynomials of degree n need not have degree n. EXAMPLE 8. The set of all functions continuous on a given interval. If the interval is[a, b], we denote this space by C(a, b). EXAMPLE 9. The set of all functions differentiable at a given point. EXAMPLE 10. The set of all functions integrable on a given interval. EXAMPLE 11. The set of all functions f defined at 1 with f(1) = 0. The number 0 isessential in this example. If we replace 0 by a nonzero number c, we violate the closureaxioms. EX A M P L E 12. The set of all solutions of a homogeneous linear differential equationy" + ay' + by = 0, where a and b are given constants. Here again 0 is essential. The setof solutions of a nonhomogeneous differential equation does not satisfy the closure axioms. These examples and many others illustrate how the linear space concept permeatesalgebra, geometry, and analysis. When a theorem is deduced from the axioms of a linearspace, we obtain, in one stroke, a result valid for each concrete example. By unifying
6 Linear spacesdiverse examples in this way we gain a deeper insight into each. Sometimes special knowl-edge of one particular example helps to anticipate or interpret results valid for otherexamples and reveals relationships which might otherwise escape notice.1.4 Elementary consequences of the axioms The following theorems are easily deduced from the axioms for a linear space. THEOREM 1.1. UNIQUENESS OF THE ZERO ELEMENT. in any linear space there is oneand only one zero element. Proof. Axiom 5 tells us that there is at least one zero element. Suppose there were two,say 0, and 0,. Taking x = OI and 0 = 0, in Axiom 5, we obtain Or + O2 = 0,.Similarly, taking x = 02 and 0 = 0,) we find 02 + 0, = 02. But Or + 02 = 02 + 0,because of the commutative law, so 0, = 02. THEOREM 1.2. UNIQUENESS OF NEGATIVE ELEMENTS. In any linear space every elementhas exactly one negative. That is, for every x there is one and only one y such that x + y = 0. Proof. Axiom 6 tells us that each x has at least one negative, namely (- 1)x. Supposex has two negatives, say y1 and yZ. Then x + yr = 0 and x + yZ = 0. Adding yZ to bothmembers of the first equation and using Axioms 5, 4, and 3, we find thatand Y2 + (x + yd = (y2 + x) + y1 = 0 + y, = y1 + 0 = y,,Therefore y1 = y2, so x has exactly one negative, the element (- 1)x. Notation. The negative of x is denoted by -x. The difference y - x is defined to bethe sum y + (-x) . The next theorem describes a number of properties which govern elementary algebraicmanipulations in a linear space. THEOREM 1.3. In a given linear space, let x and y denote arbitrary elements and let a and bdenote arbitrary scalars. Then we'have the following properties: (a) Ox = 0. (b) a0 = 0. (c) (-a)x = -(ax) = a(-x). (d) Ifax=O,theneithera=Oorx=O. ( e ) Ifax=ayanda#O, thenx=y. ( f ) Ifax=bxandx#O,thena=b. (g> -(x + y) = (-4 + C-y) = --x - y. (h) x + x = 2x, x + x +x = 3x, andingeneral, &x = nx.
8 Linear spaces28. All vectors in V,, that are linear combinations of two given vectors A and B.29. Let V = R+, the set of positive real numbers. Define the "sum" of two elements x and y in V to be their product x y (in the usual sense), and define "multiplication" of an element x in V by a scalar c to be xc. Prove that V is a real linear space with 1 as the zero element.30. (a) Prove that Axiom 10 can be deduced from the other axioms. (b) Prove that Axiom 10 cannot be deduced from the other axioms if Axiom 6 is replaced by Axiom 6': For every x in V there is an element y in V such that x + y = 0.3 1. Let S be the set of all ordered pairs (x1, xZ) of real numbers. In each case determine whether or not S is a linear space with the operations of addition and multiplication by scalars defined as indicated. If the set is not a linear space, indicate which axioms are violated. (4 (x1,x2) + (y19y2) = (x1 +y1,x2 +y,), 4x1, x2) = @Xl) 0). (b) (-99x2) + (y1,y,) = (~1 +yl,O), 4X1,X,) = (ax,, ax,>. cc> (Xl, x2) + cy1,y2> = (Xl, x2 +y2>9 4x1, x2> = (~17 QX2>. (4 @1,x2) + (yl,y2) = (Ix, + x,l,ly1 +y,l)t 4x1, x2) = (lql, lq!l) f32. Prove parts (d) through (h) of Theorem 1.3.1.6 Subspaces of a linear space Given a linear space V, let S be a nonempty subset of V. If S is also a linear space, withthe same operations of addition and multiplication by scalars, then S is called a subspaceof V. The next theorem gives a simple criterion for determining whether or not a subset ofa linear space is a subspace. THEOREM 1.4. Let S be a nonempty subset of a linear space V. Then S is a subspaceif and only if S satisfies the closure axioms. Proof. If S is a subspace, it satisfies all the axioms for a linear space, and hence, inparticular, it satisfies the closure axioms. Now we show that if S satisfies the closure axioms it satisfies the others as well. Thecommutative and associative laws for addition (Axioms 3 and 4) and the axioms formultiplication by scalars (Axioms 7 through 10) are automatically satisfied in S becausethey hold for all elements of V. It remains to verify Axioms 5 and 6, the existence of a zeroelement in S, and the existence of a negative for each element in S. Let x be any element of S. (S has at least one element since S is not empty.) By Axiom2, ax is in S for every scalar a. Taking a = 0, it follows that Ox is in S. But Ox = 0, byTheorem 1.3(a), so 0 E S, and Axiom 5 is satisfied. Taking a = - 1, we see that (-1)xis in S. But x + (- 1)x = 0 since both x and (- 1)x are in V, so Axiom 6 is satisfied inS. Therefore S is a subspace of V. DEFINITION. Let S be a nonempty subset of a linear space V. An element x in V of theform k x = z: cixi ) i=lwhere x1,. . . , xk are all in S and cl, . . . , ck are scalars, is called a$nite linear combinationof elements of S. The set of alljnite linear combinations of elements of S satisjies the closureaxioms and hence is a subspace of V. We call this the subspace spanned by S, or the linearspan of S, and denote it by L(S). If S is empty, we dejne L(S) to be {0}, the set consistingof the zero element alone.
Dependent and independent sets in a linear space 9 Different sets may span the same subspace. For example, the space V, is spanned byeach of the following sets of vectors: {i,j}, {i,j, i +j}, (0, i, -i,j, -j, i + j}. The spaceof all polynomialsp(t) of degree < n is spanned by the set of n + 1 polynomials (1, t, t2, . . . ) P}.It is also spanned by the set {I, t/2, t2/3, . . . , t"/(n + l>>, and by (1, (1 + t), (1 + t)", . . . ,(1 + t)"}. The space of all polynomials is spanned by the infinite set of polynomials(1, t, t2, . . .}. A number of questions arise naturally at this point. For example, which spaces can bespanned by a finite set of elements? If a space can be spanned by a finite set of elements,what is the smallest number of elements required? To discuss these and related questions,we introduce the concepts of dependence, independence, bases, and dimension. These ideaswere encountered in Volume I in our study of the vector space V, . Now we extend themto general linear spaces.1.7 Dependent and independent sets in a linear space DEFINITION. A set S of elements in a linear space V is called dependent if there is a-finiteset of distinct elements in S, say x1, . . . , xg, and a corresponding set of scalars cl, . . . , c,,not all zero, such thatAn equation 2 c,x( = 0 with not all ci = 0 is said to be a nontrivial representation of 0.The set S is called independent ifit is not dependent. In this case, for all choices of distinctelements x1, . . . , xk in S and scalars cl, . . . , ck, ii cixi = O implies c1=c2=..*=ck=o. Although dependence and independence are properties of sets of elements, we also applythese terms to the elements themselves. For example, the elements in an independent setare called independent elements. If S is a finite set, the foregoing definition agrees with that given in Volume I for thespace V,. However, the present definition is not restricted to finite sets. EXAMPLE 1. If a subset T of a set S is dependent, then S itself is dependent. This islogically equivalent to the statement that every subset of an independent set is independent. EXPMPLE 2. If one element in S is a scalar multiple of another, then S is dependent. EXAMPLE 3. If 0 E S, then S is dependent. EXAMPLE 4. The empty set is independent,
10 Linear spaces Many examples of dependent and independent sets of vectors in V,, were discussed inVolume I. The following examples illustrate these concepts in function spaces. In eachcase the underlying linear space V is the set of all real-valued functions defined on the realline. EXAMPLE 5. Let ui(t) = co? t , uz(t) = sin2 t , us(f) = 1 for all real t. The Pythagoreanidentity shows that u1 + u2 - uQ = 0, so the three functions ui, u2, u, are dependent. EXAMPLE 6. Let uk(t) = tk for k = 0, 1,2. . . . , and t real. The set S = {u,, , ui, u2, . . .}is independent. To prove this, it suffices to show that for each n the n + 1 polynomialsu,, 4, *. * 3 u, are independent. A relation of the form 1 c,u, = 0 means that(1.1) -&tk = 0 k=Ofor all real 1. When t = 0, this gives co = 0 . Differentiating (1.1) and setting t = 0,we find that c1 = 0. Repeating the process, we find that each coefficient ck is zero. EXAMPLE 7. If a,,..., a, are distinct real numbers, the n exponential functions q(x) = ea@, . . . , u,(x) = eanrare independent. We can prove this by induction on n. The result holds trivially whenn = 1 . Therefore, assume it is true for n - 1 exponential functions and consider scalarsCl, * *. 7 c, such that(1.2) tckeakx = 0. k=lLet alIf be the largest of the n numbers a,, . . . , a,. Multiplying both members of (1.2)by e- a~x, we obtainIf k # M, the number ak - aAl is negative. Therefore, when x + + co in Equation (1.3),each term with k # M tends to zero and we find that cnl = 0. Deleting the Mth term from(1.2) and applying the induction hypothesis, we find that each of the remaining n - 1coefficients c, is zero. THEOREM 1.5. Let s = {X1,..., xk} be an independent set consisting of k elements in alinear space V and let L(S) be the subspace spanned by S. Then every set of k + 1 elementsin L(S) is dependent. Proof. The proof is by induction on k, the number of elements in S. First supposek = 1. Then, by hypothesis, S consists of one element xi, where x1 # 0 since S isindependent. Now take any two distinct elements y1 and yZ in L(S). Then each is a scalar
Dependent and independent sets in a linear space 11multiple of x1, say y1 = clxl and yZ = cZxl, where c, and c2 are not both 0. Multiplyingy1 by c2 and y, by c1 and subtracting, we find that c2.h - c,y, = 0.This is a nontrivial representation of 0 soy, and y2 are dependent. This proves the theoremwhen k = 1 . Now we assume the theorem is true for k - 1 and prove that it is also true for k. Takeany set of k + 1 elements in L(S), say T = {yr , yZ, . . I , yk+r} . We wish to prove that Tisdependent. Since each yi is in L(S) we may write k(1.4) yi = 2 a, jxj i-1f o r e a c h i = 1,2,... , k + 1 . We examine all the scalars ai, that multiply x1 and split theproof into two cases according to whether all these scalars are 0 or not. CASE 1. ai, = 0 for every i = 1,2, . . . , k + 1 . In this case the sum in (1.4) does notinvolve x1, so each yi in T is in the linear span of the set S' = {x2, . . . , xk} . But S' isindependent and consists of k - 1 elements. By the induction hypothesis, the theorem istrue for k - 1 so the set T is dependent. This proves the theorem in Case 1. CASE 2. Not all the scaIars ai, are zero. Let us assume that a,, # 0. (If necessary, wecan renumber the y's to achieve this.) Taking i = 1 in Equation (1.4) and multiplying bothmembers by ci, where ci = ail/all, we get k Ciyl = ailxl + 1 CiUl jXj . j=2From this we subtract Equation (1.4) to get k C,yl - yi = x(Cial j - aij>xj 3 j=2fori=2,..., k + 1 . This equation expresses each of the k elements ciy, - yi as a linearcombination of the k - 1 independent elements .x2, . . . , xk . By the induction hypothesis,the k elements ciy, - yi must be dependent. Hence, for some choice of scalars t,, . . . ,tk+l, not all zero, we have kfl iz2ti(ciYl - Yi) = O 9from which we findBut this is a nontrivial linear combination of y,, . . . , yh.+l which represents the zero ele-ment, so the elements y1 , . . . , yri.r must be dependent. This completes the proof.
12 Linear spaces1.8 Bases and dimension DEFINITION. A jinite set S of elements in a linear space V is called aJnite basis .for Vif S is independent and spans V. The space V is called$nite-dimensional if it has a jinitebasis, or if V consists of 0 alone. Otherwise, V is called injinite-dimensional. THEOREM 1.6. Let V be a jnite-dimensional linear space. Then every jnite basis for Vhas the same number of elements. Proof. Let S and T be two finite bases for V. Suppose S consists of k elements and Tconsists of m elements. Since S is independent and spans V, Theorem 1.5 tells us thatevery set of k + 1 elements in Vis dependent. Therefore, every set of more thank elementsin V is dependent. Since T is an independent set, we must have m 5 k. The same argu-ment with S and T interchanged shows that k < m . Therefore k = m . DEFINITION. If a linear space V has a basis of n elements, the integer n is called thedimension of V. We write n = dim V. If V = {O}!, we say V has dimension 0. EXAMPLE 1. The space V, has dimension n. One basis is the set of n unit coordinatevectors. EXAMPLE 2. The space of all polynomials p(t) of degree < n has dimension n + 1 . Onebasis is the set of n + 1 polynomials (1, t, t2, . . . , t'"}. Every polynomial of degree 5 n is alinear combination of these n + 1 polynomials. EXAMPLE 3. The space of solutions of the differential equation y" - 2y' - 3y = 0 hasdimension 2. One basis consists of the two functions ul(x) = e-", u2(x) = e3x, Everysolution is a linear combination of these two. EXAMPLE 4. The space of all polynomials p(t) is infinite-dimensional. Although theinfinite set (1, t, t2, . . .} spans this space, no$nite set of polynomials spans the space. THEOREM 1.7. Let V be a jinite-dimensional linear space with dim V = n. Then wehave the following: (a) Any set of independent elements in V is a s&set of some basis for V. (b) Any set of n independent elements is a basisf;pr V. Proof. To prove (a), let S = {x1, . . . , xk} be any independent set of elements in V.If L(S) = V, then S is a basis. If not, then there is some element y in V which is not inL(S). Adjoin this element to S and consider the new set S' = {x1, . . . , xk, y} . If thisset were dependent there would be scalars cl, . . . , c~+~, not all zero, such that izlCiXi + cktly = 0 *But Ck+l # 0 since xi, . . . , xk are independent. Hence, we could solve this equation fory and find that y E L(S), contradicting the fact that y is not in L(S). Therefore, the set S'
Exercises 13is independent but contains k + 1 elements. If L(S') = V, then S' is a basis and, since Sis a subset of S', part (a) is proved. If S' is not a basis, we can argue with S' as we didwith S, getting a new set S" which contains k + 2 elements and is independent. If S" is abasis, then part (a) is proved. If not, we repeat the process. We must arrive at a basis ina finite number of steps, otherwise we would eventually obtain an independent set withit + 1 elements, contradicting Theorem 1.5. Therefore part (a) is proved. To prove (b), let S be any independent set consisting of II elements. By part (a), S is asubset of some basis, say B. But by Theorem 1.6, the basis B has exactly n elements, soS= B.1.9 Components Let V be a linear space of dimension II and consider a basis whose elements e, , . . . , e,are taken in a given order. We denote such an ordered basis as an n-tuple (e,, . . . , e,).If x E V, we can express x as a linear combination of these basis elements:(1.5) x = $ ciei . i=lThe coefficients in this equation determine an n-tuple of numbers (c,, . . . , CJ that isuniquely determined by x. In fact, if we have another representation of x as a linearcombination of e,, . . . , e,, say x = I7z1 d,e,, then by subtraction from (1.5), we find that& (ci - d,)e, = 0. Bu t since the basis elements are independent, this implies ci = diforeachi,sowehave(c, ,..., c,)=(d, ,..., d,). The components of the ordered n-tuple (c,, . . . , CJ determined by Equation (1.5) arecalled the components of x relative to the ordered basis (e, , . . . , e,).1.10 Exercises In each of Exercises 1 through 10, let S denote the set of all vectors (x, y, z) in V3 whose com-ponents satisfy the condition given. Determine whether S is a subspace of V3. If S is a subspace,compute dim S. 1. X = 0. 6.x=y o r x=z. 2. x +y =o. 7. x2-y2=o. 3.x+y+z=o. 8. x fy = 1. 4. x =y. 9..y=2x a n d z=3x. 5 . x =y =z. 10. x + J + z = 0 and x - y - z = 0. Let P, denote the linear space of all real polynomials of degree < it, where n is fixed. In eachof Exercises 11 through 20, let S denote the set of all polynomials f in P, satisfying the conditiongiven. Determine whether or not S is a subspace of P, . If S is a subspace, compute dim S.11. f(0) = 0. 16. f(0) = f(2) .12. j-'(O) = 0. 17. f is even.13. j-"(O) = 0. 18. f is odd.14. f(O) +f'(o> = 0. 19. f has degree _< k, where k < n, or f = 0.15. f(0) =f(l). 20. f has degree k, where k < n , or f = 0.21. In the linear space of all real polynomials p(t), describe the subspace spanned by each of the following subsets of polynomials and determine the dimension of this subspace. 6-4 (1, t2, t4>; (b) {t, t3, t5>; cc> 0, t2> ; (d) { 1 + t, (1 + t,"}.
14 Linear spaces22. In this exercise, L(S) denotes the subspace spanned by a subset S of a linear space V. Prove each of the statements (a) through (f). (a) S G L(S). (b) If S G T G V and if T is a subspace of V, then L(S) c T. This property is described by saying that L(S) is the smallest subspace of V which contains 5'. (c) A subset S of V is a subspace of V if and only if L(S) = S. (d) If S c T c V, then L(S) c L(T). (e) If S and Tare subspaces of V, then so is S n T. (f) If S and Tare subsets of V, then L(S n T) E L(S) n L(T). (g) Give an example in which L(S n T) # L(S) ~-1 L(T).23. Let V be the linear space consisting of all real-valued functions defined on the real line. Determine whether each of the following subsets of V is dependent or independent. Compute the dimension of the subspace spanned by each set. ,ea2,ebz},a #b. (f) {cos x, sin x>. I%: i' ear, xeax}. (g) {cosz x, sin2 x}. iz il, eaz, xeaz). (h) {'I, cos 2x, sin2 x}. eax, xeax, x2eax}. (i) {sin x, sin 2x}. (e) {e", ec", cash x}. (j) {e" cos x, eP sin x}.24. Let V be a finite-dimensional linear space, and let S be a subspace of V. Prove each of the following statements. (a) S is finite dimensional and dim S 2 dim V. (b) dim S = dim V if and only if S = V. (c) Every basis for S is part of a basis for V. (d) A basis for V need not contain a basis for S.1.11 Inner products, Euclidean spaces. Norms In ordinary Euclidean geometry, those properties that rely on the possibility of measuringlengths of line segments and angles between lines are called metric properties. In our studyof V,, we defined lengths and angles in terms of the dot product. Now we wish to extendthese ideas to more general linear spaces. We shall introduce first a generalization of thedot product, which we call an inner product, and then define length and angle in terms of theinner product. The dot product x *y of two vectors x = (x1, . . . , x,) and y = (ul, . . . , yn) in V, wasdefined in Volume I by the formula(1.6) x * y = i x,y,. i=IIn a general linear space, we write (x, JJ) instead of x * y for inner products, and we definethe product axiomatically rather than by a specific formula. That is, we state a number ofproperties we wish inner products to satisfy and we regard these properties as axioms. DEFINITION. A real linear space V is said to have an inner product if for each pair ofelements x and y in V there corresponds a unique real number (x, y) satisfying the followingaxioms for all choices of x, y, z in V and all real scalars c. (1) (XT y> = oi, 4 (commutativity, or symmetry). (2) (x, y + z> = (x, y> + (x3 z> (distributivity, or linearity). (3) 4x2 .Y> = (cx, Y> (associativity, or homogeneity). (4) (x3 x> > 0 if x#O (positivity).
Inner products, Euclidean spaces. Norms 15A real linear space with an inner product is called a real Euclidean space. Note: Taking c = 0 in (3), we find that (0,~) = 0 for all y. In a complex linear space, an inner product (x, y) is a complex number satisfying thesame axioms as those for a real inner product, except that the symmetry axiom is replacedby the relation(1') (X>Y> = (YP 4, (Hermitian? symmetry)where (y, x) denotes the complex conjugate of (y, x). In the homogeneity axiom, the scalarmultiplier c can be any complex number. From the homogeneity axiom and (l'), we getthe companion relation -_ _(3') (x, cy) = (cy, x) = Q, x) = qx, y). A complex linear space with an inner product is called a complex Euclidean 'space.(Sometimes the term unitary space is also used.) One example is complex vector spaceV,(C) discussed briefly in Section 12.16 of Volume I. Although we are interested primarily in examples of real Euclidean spaces, the theoremsof this chapter are valid for complex Euclidean spaces as well. When we use the termEuclidean space without further designation, it is to be understood that the space can bereal or complex. The reader should verify that each of the following satisfies all the axioms for an innerproduct. EXAMPLE 1. In I', let (x, y) = x . y , the usual dot product of x and y. EXAMPLE 2. If x = (xi, XJ and y = (yi , yJ are any two vectors in V,, define (x, y) bythe formula (x3 Y) = %Yl + XlY2 + X2Yl + X2Y2 *This example shows that there may be more than one inner product in a given linear space. EXAMPLE 3. Let C(a, b) denote the linear space of all real-valued functions continuouson an interval [a, b]. Define an inner product of two functions f and g by the formula CL d = jab J-(&At) dt .This formula is analogous to Equation (1.6) which defines the dot product of two vectorsi n I!,. The function values f(t) and g(t) play the role of the components xi and yi , andintegration takes the place of summation.t In honor of Charles Hermite (1822-1901), a French mathematician who made many contributions toalgebra and analysis.
16 Linear spaces EXAMPLE 4. In the space C(a, b), define u-3 d = jab W(W(Od~) dt,where w is a fixed positive function in C(a, b). The function w is called a weightfunction.In Example 3 we have w(t) = 1 for all t. EXAMPLE 5. In the linear space of all real polynomials, define CL d = jam e-tf(MO dt.Because of the exponential factor, this improper integral converges for every choice ofpolynomials /and g. THEOREM 1.8. In a Euclidean space V, every inner product satisfies the Cauchy-Schwarzinequality: I(x,y)12 5 (x, x)(y, y) for all x andy in V.Moreover, the equality sign holds lyand only if x and y are dependent. Proof. If either x = 0 or y = 0 the result holds trivially, so we can assume that bothx and y are nonzero. Let z = ax + by, where a and b are scalars to be specified later. Wehave the inequality (z, z) >_ 0 for all a and b. When we express this inequality in terms of xand y with an appropriate choice of a and b we will obtain the Cauchy-Schwarz inequality. To express (z, z) in terms of x and y we use properties (l'), (2) and (3') to obtain (z,Z> = (ax + by, ax + by) = (ax, ax) + (ax, by) + (by, ax) + (by, by) = a@, x> + a&x, y) + bii(y, x) + b&y, y) 2 0.Taking a = (y, y) and cancelling the positive factor (J, y) in the inequality we obtain 01, y>(x, 4 + 6(x, y> + Ny, xl + b6 2 0.Now we take b = -(x, y) . Then 6 = - (y, x) and the last inequality simplifies to (Y, y)(x, x) 2 (x, y>c.Y9 x> = I(& yv.This proves the Cauchy-Schwarz inequality. The equality sign holds throughout the proofif and only if z = 0. This holds, in turn, if and only if x and y are dependent. EXAMPLE. Applying Theorem 1.8 to the space C(a, b) with the inner product (f, g) =j,bf(t)g(t) dt , we find that the Cauchy-Schwarz inequality becomes (jbf(MO a dt) I (jabfZW dt)( jab g"(t) dl).
Inner products, Euclidean spaces. Norms 17 The inner product can be used to introduce the metric concept of length in any Euclideanspace. DEFINITION. In a Euclidean space V, the nonnegative number IIx I/ deJned by the equation llxjl = (x, x)"is called the norm of the element x. When the Cauchy-Schwarz inequality is expressed in terms of norms, it becomes IGGY)I 5 llxll M . Since it may be possible to define an inner product in many different ways, the normof an element will depend on the choice of inner product. This lack of uniqueness is to beexpected. It is analogous to the fact that we can assign different numbers to measure thelength of a given line segment, depending on the choice of scale or unit of measurement.The next theorem gives fundamental properties of norms that do not depend on the choice of inner product. THEOREM 1.9. In a Euclidean space, every norm has the following properties for allelements x and y and all scalars c: (4 II-4 = 0 if x=0. @I II4 > 0 if x#O (positivity). cc> Ilcxll = IcIll4 (homogeneity). (4 Ilx + YII I l/x/I + Ilyll (triangle inequality). The equality sign holds in (d) if x = 0, ify = 0, or if y = cxfor some c > 0. Proof. Properties (a), (b) and (c) follow at once from the axioms for an inner product.To prove (d), we note that Il.~+yl12=(~+y,~+y>=~~,~~+~y,y>+~~,y>+cv,~> = lIxl12 + llyl12 + (x3 y> + t-x, y>.The sum (x, y) + (x, y) is real. The Cauchy-Schwarz inequality shows that 1(x, y)l 5II-4 llyll and IGG y)I I II4 llyll , so we have /lx + yl12 I lIxl12 + llYl12 + 2llxll llyll = Wll + llyll>".This proves (d). When y = cx , where c > 0, we have /lx +yII = IIX + cxll = (1 + c> IL-II = llxll + IICXII = II4 + llyll .
18 Linear spaces DEFINITION. In a real Euclidean space V, the angle between two nonzero elements x andy is dejned to be that number t9 in the interval 0 5 8 < TT which satisfies the equation cos e = - (x9 Y)(1.7) IIXII llvll ' Note: The Cauchy-Schwarz inequality shows that the quotient on the right of (1.7) lies in the interval [ - 1 , 11, so there is exactly one 0 in [0, 7~1 whose cosine is equal to this quotient.1.12 Orthogonality in a Euclidean space DEFINITION. In a Euclidean space V, two elements x and y are called orthogonal if theirinner product is zero. A subset S of V is calIed an orthogonal set if (x, y) = 0 for every pairof distinct elements x and y in S. An orthogonal set is called orthonormal if each of itselements has norm 1. The zero element is orthogonal to every element of V; it is the only element orthogonal toitself. The next theorem shows a relation between orthogonality and independence. THEOREM 1.10. In a Euclidean space V, every orthogonal set of nonzero elements isindependent. In particular, in a jinite-dimensional Euclidean space with dim V = n, everyorthogonal set consisting of n nonzero elements is a basis for V. Proof. Let S be an orthogonal set of nonzero elements in V, and suppose some finitelinear combination of elements of S is zero, saywhere each xi E S. Taking the inner product of each member with x1 and using the factthat (xi , xi) = 0 if i # 1 , we find that cl(xl, x1) = 0. But (x1, x,) # 0 since xi # 0 soc1 = 0. Repeating the argument with x1 replaced by xi, we find that each cj = 0. Thisproves that S is independent. If dim V = n and if S consists of n elements, Theorem 1.7(b)shows that S is a basis for V. EXAMPLE. In the real linear space C(O,27r) with the inner product (f, g) = JiBf(x)g(x) dx,let S be the set of trigonometric functions {u,, ul, u2, . . .} given by %&4 = 1, uznpl(x) = cos nx, uZn(x) = sin nx, f o r n = 1,2,....If m # n, we have the orthogonality relations 2n u~(x)u,(x) dx = 0, s0
Orthogonality in a Euclidean space 19so S is an orthogonal set. Since no member of S is the zero element, S is independent. Thenorm of each element of S is easily calculated. We have (u,, uO) = j'$' dx = 27r and, forn 2 1, we have (~1~~~~) u2,r-l 1 -wo cos2 nx dx = T, - i"" (uzT1, uzvr) =Ib?" sin2 nx dx = T.Therefore, iluOll = J% and lIu,/l = V% for n 2 1 . Dividing each zd, by its norm, weobtain an orthonormal set {pO, cpi , yz, . . .} where ~j~ = u,/~Iu,lI Thus, we have q+)(x) = 1 cp2,Ax) = y > ql,,(x) = s= ) f o r n>l J2n ' 'G In Section 1.14 we shall prove that every finite-dimensional Euclidean space has anorthogonal basis. The next theorem shows how to compute the components of an elementrelative to such a basis. THEOREM I .l 1. Let V he a finite-dimerwionai Euclidean space with dimension n, andwume that S = {El, . . . , e,>IS an orthogonal basis,fbr V. [fan element x is expressed asg linear combination of the basis elements, say:1.8) .x = f c,e, , c=lthen its components relative to the ordered basis (e, , . . . , e,> are given by the,formulas'1.9)rn particular, if S is an orthonormal basis, each cj is given by11.10) cj = (x, ej) . Proof. Taking the inner product of each member of (1.8) with ej, we obtain (X, ej) = i c,(ei, ej) = c,(ej, ej) i=-;* Ante (ei, eJ = 0 if i #j. This implies (1.9), and when (ej, ej) = 1, we obtain (1.10). If {e,, . . . , e,} is an orthonormal basis, Equation (1.9) can be written in the form 1.11) X = f (x, ei)ei . i=l The next theorem shows that in a finite-dimensional Euclidean space with an orthonormal oasis the inner product of two elements can be computed in terms of their components.
20 Linear spaces THEOREM 1.12. Let V be a$nite-dimensional Euclidean space of dimension n, and assumefhat {e,, . . . , e,} is an orthonormal basis for V. Then for every puir of elements x and y in V,we have(1.12) (Parseval's formula).In particular, when x = y , we have Proof, Taking the inner product of both members of Equation (1.11) withy and usingthe linearity property of the inner product, we obtain (1.12). When x = y, Equation(1.12) reduces to (1.13). Note: Equation (1.12) is named in honor of hf. A. ParsevaI (circa 1776-1836), who obtained this type of formula in a special function space. Equation (1.13) is a generalization of the theorem of Pythagoras.1.13 Exercises 1. Let x = (x1, . . . , x,) andy = (yl, . . . , yn) be arbitrary vectors in V, . In each case, determine whether (x, y) is an inner product for V,, if (x, y) is defined by the formula given. In case (x, y) is not an inner product, tell which axioms are not satisfied. (4 (A y> = 5 xi lyil (4 (x, y) = ( i&:yf)"2 . i=l (4 0, y> = j$ (xi + yd2 - t$lx1 - $IIK. 2. Suppose we retain the first three axioms for a real inner product (symmetry, linearity, and homogeneity but replace the fourth axiom by a new axiom (4'): (x, x) = 0 if and only if x = 0. Prove that either (x, x) > 0 for all x # 0 or else (x, x) < 0 for all x # 0. [Hint: Assume (x, x) > 0 for some x # 0 and (y, y) < 0 for some y # 0. In the space spanned by {x, y), find an element z # 0 with (z, z) = 0.1 Prove that each of the statements in Exercises 3 through 7 is valid for all elements x and y in areal Euclidean space. 3. (x, y) = 0 if and only if /Ix + yll = l/x - yl/ . 4. (x, y) = 0 if and only if 11x + yj12 = j/x112 + 11~11~. 5. (x, y) = 0 if and only if 11x + cyll > ]jxll for all real c. 6. (x + y, x - y) = 0 if and only if (Ix/I = liyjj. 7. If x and y are nonzero elements making an angle 0 with each other, then IIX - yl12 = llxl12 + Ilyl12 - 2 IIXII llyll cos 0.
22 Linear spaces (a) Prove that the integral for (f, g) converges absolutely for each pair of functions f and g in V. [Hint: Use the Cauchy-Schwarz inequality to estimate the integral jf e-t 1f (t)g(t)l dt.] (b) Prove that V is a linear space with (f, g) as an inner product. (c) Compute (f,g) iff(t) = e& and&t) = P, where n = 0, 1,2, . . . .15. In a complex Euclidean space, prove that the inner product has the following properties for all elements X, y and z, and all complex a and b. (4 (ax, by) = &x, y). (b) (x, ay + bz) = rf(x, y) + 6(x, z).16. Prove that the following identities are valid in every Euclidean space. (a) Ilx +yl12 = l/xl12 + llyl12 + (x,y> + (y, x). (b) I/x + yl12 - lb - yl12 = 2(x, y) + xy, 4. (4 l/x + yl12 + lx - yl12 = 2 llxl12 + 2 IIy112.17. Prove that the space of all complex-valued functions continuous on an interval [a, b] becomes a unitary space if we define an inner product by the formula (fvg) = s: Wf(QgO4 where w is a fixed positive function, continuous on [a, b].1.14 Construction of orthogonal sets. The Gram-Scltmidt process Every finite-dimensional linear space has a finite basis. If the space is Euclidean, we canalways construct an orthogonal basis. This result will be deduced as a consequence of ageneral theorem whose proof shows how to construct orthogonal sets in any Euclideanspace, finite or infinite dimensional. The construction is called the Gram-Schmidt orthog-onalizationprocess, in honor of J. P. Gram (1850-1916) and E. Schmidt (18451921). THEOREM 1.13. ORTHOGONALIZATION THEOREM . Let x1,x2,. .., be ajinite or intnitesequence of elements in a Euclidean space V, and let L(x,, . . . , xk) denote the subspacespanned by thejrst k of these elements. Then there is a corresponding sequence of elementsy1,y2, * * * 9 in V which has the following properties for each integer k: (a) The element yr is orthogonal to every element in the subspace L(yl, . . . , yk-J. (b) The subspace spanned by yl, . . . , yk is the same as that spanned by x1, . . . , x, : uyl,. . . ,yJ = L(x,, . . . , XTJ. (c) The sequence yl, y, , . . . , is unique, except for scalar factors. That is, ifyi , y: , . . . , isanother sequence of elements in V satisfying properties (a) and (b) for all k, then for each kthere is a scalar ck such that y; = ckyr . Proof. We construct the elements yr, y2, . . . , by induction. To start the process, wetake yr = x1. Now assume we have constructed yl, . . . , y,. so that (a) and (b) are satisfiedwhen k = r . Then we define y,.+r by the equation(1.14) Yr+1 = Xr+l - & ad+ y
Construction of orthogonal sets. The Gram-Schmidt process 23where the scalars a,, . . . , a, are to be determined. For j < r, the inner product of yI+rwith yj is given bysince (yi, yj) = 0 if i #j . If yj # 0, we can make yr+r orthogonal to yj by taking(1.15) a _ (XT+1 7 .Yi) 3 (Yj, Yi) 'If yj = 0, then yr+i is orthogonal to yj for any choice of aj, and in this case we chooseaj=O. Thus, the element Y?+~ is well defined and is orthogonal to each of the earlierelements yr , . . . , y, . Therefore, it is orthogonal to every element in the subspaceThis proves (a) when k = r + 1. To prove (b) when k = r + 1, we must show that L(y,, . . . , y,.+J = L(x, , . . . , x,+r),given that L(y,, . . . , yr) = L(x,, . . . , x,) . The first r elements yl, . . . , y,. are inand hence they are in the larger subspace L(x, , . . . , x,+~). The new element yrsl given by(1.14) is a difference of two elements in ,5(x,, . . , , x,+~) so it, too, is in L(x,, . . . , x,+r).This proves thatEquation (1.14) shows that x,+i is the sum of two elements in LQ, , . . . , yr+r) so a similarargument gives the inclusion in the other direction: UXl, . . . 9 x,+1) s uyl, . . . ,y7+1).This proves (b) when k = r + 1. Therefore both (a) and (b) are proved by induction on k. Finally we prove (c) by induction on k. The case k = 1 is trivial. Therefore, assume (c)is true for k = r and consider the element y:+r . Because of (b), this element is inso we can write Yk+* =;ciyi = ZT + Cr+lYr+l,where z, E L(y,, . . . , y,.) . We wish to prove that z, = 0. By property (a), both vi+,, and~,+ry,.+~ are orthogonal to z, . Therefore, their difference, z,, is orthogonal to z, . In otherwords, z, is orthogonal to itself, so z, = 0. This completes the proof of the orthogonaliza-tion theorem.
24 Linear spaces In the foregoing construction, suppose we have Y?,.~ = 0 for some r. Then (1.14)shows that x,+~ is a linear combination of yl, . . . ,y,, and hence of x1, . . . , xc, so theelements x1, . . . , x,.,, are dependent. In other words, if the first k elements x1,. . . , x,are independent, then the corresponding elements y1 , . . . , yk are nonzero. In this case thecoefficients ai in (1.14) are given by (1.15), and the formulas defining y, , . . . , yk become(1.16) y, = x1, Yr+l = %+1 - for r = 1,2, . . . , k - 1.These formulas describe the Gram-Schmidt process for constructing an orthogonal set ofnonzero elements yl, . . . , y, which spans the same subspace as a given independent setXl,. . . ,x,. In particular, if x1, . . . , x, is a basis for a finite-dimensional Euclidean space,theny,, . . . ,yk is an orthogonal basis for the same space. We can also convert this to anorthonormal basis by normalizing each element yi, that is, by dividing it by its norm.Therefore, as a corollary of Theorem 1.13 we have the following. THEOREM 1.14. Every$nite-dimensional Euclidean space has an orthonormal basis. If x and y are elements in a Euclidean space, withy # 0, the elementis called the projection of x along y. In the Gram-Schmidt process (1.16), we constructthe element Y,.+~ by subtracting from x,.+~ the projection of x,+r along each of the earlierelements yl, . . . , yr. Figure 1.1 illustrates the construction geometrically in the vectorspace V,. FrooaE 1.1 The Gram-Schmidt process in V3 . An orthogonal set {yl, yZ , y3} is constructed from a given independent set {x1, x2, ~3.
26 Linear spacesWe shall encounter these polynomials again in Chapter 6 in our further study of differentialequations, and we shall prove that n! d" - - (t2 - 1)". yn(f) = (2n)! dt"The polynomials P, given by (2n)! p?st> = - y,(t) = 2"(n !)" h -$ 0" - 1)"are known as the Legendrepolynomials. The polynomials in the corresponding orthonormalsequence fro, ply v2, . . . , given by ~7~ = y,/llynll are called the normalized Legendre poly-nomials. From the formulas for yo, . . . , y5 given above, we find that f&l(t) = Ji ) q%(t) = 4 t ) q*(t) = &JS (3t2 - 1)) Q)3(t) = &J3 (5t3 - 3t), p4(t) = +Ji (35th - 30t* + 3), p5(t) = $& (63t" - 70t3 + 19).1.15. Orthogonal complements. Projections Let V be a Euclidean space and let S be a finite-dimensional subspace. We wish toconsider the following type of approximation problem: Given an element x in V, to deter-mine an element in S whose distance from x is as small as possible. The distance betweentwo elements x and y is defined to be the norm 11x - yII . Before discussing this problem in its general form, we consider a special case, illustratedin Figure 1.2. Here V is the vector space V, and S is a two-dimensional subspace, a planethrough the origin. Given x in V, the problem is to find, in the plane S, that point snearest to x. If x E S, then clearly s = x is the solution. If x is not in S, then the nearest point sis obtained by dropping a perpendicular from x to the plane. This simple example suggestsan approach to the general approximation problem and motivates the discussion thatfollows. DEFINITION. Let S be a subset of a Euclidean space V. An element in V is said to beorthogonal to S if it is orthogonal to every element of S. The set of' all elements orthogonalto S is denoted by S-' and is called "S perpendicular." It is a simple exercise to verify that Sl is a subspace of V, whether or not S itself is one.In case S is a subspace, then S1 is called the orthogonal complement of S. EXAMPLE. If S is a plane through the origin, as shown in Figure 1.2, then S1 is a linethrough the origin perpendicular to this plane. This example also gives a geometric inter-pretation for the next theorem.
Orthogonal complements. Projections 27 F IGURE 1.2 Geometric interpretation of the orthogonal decomposition theorem in V,. THEOREM 1.15. O R T H OGONAL DECOMPOSITION THEOREM . Let V be a Euclidean spaceand let S be ajnite-dimensional subspace of V. Then every element x in V can be representeduniquely as a sum of two elements, one in S and one in Sl. That is, we have(1.17) x=s+s', where s E S and d- E 5-l.Moreover, the norm of x is given by the Pythagorean formula(1.18) llxl12 = IIsl12 + Il~1112. Proof. First we prove that an orthogonal decomposition (1.17) actually exists. SinceS is finite-dimensional, it has a finite orthonormal basis, say {e, , . . . , e,}. Given x, definethe elements s and sL as follows:(1.19) s = i (x, ei)ei, SI = x - s . i=lNote that each term (x, e,)e, is the projection of x along e, . The element s is the sum of theprojections of x along each basis element. Since s is a linear combination of the basiselements, s lies in S. The definition of & shows that Equation (1.17) holds. To prove thatd lies in Sl, we consider the inner product of sL and any basis element ej . We have (S', ej> = (x - s, ej) = (x, e,) - (s, ei) .But from (1.19), we find that (s, eJ = (x, e,), so sL is orthogonal to ei. Therefore sLis orthogonal to every element in S, which means that sL E SL . Next we prove that the orthogonal decomposition (1.17) is unique. Suppose that xhas two such representations, say(1.20) x=s+sl- and x=t+tl,
28 Linear spaceswhere s and t are in S, and sL and t' are in S I. We wish to prove that s = t and sL = t1 .From (1.20), we have s - t = t1 - sL, so we need only prove that s - I = 0. Buts - t E Sand t1 - s1 E SL so s - t is both orthogonal to tl - .sL and equal to t1 - & .Since the zero element is the only element orthogonal to itself, we must have s - t = 0.This shows that the decomposition is unique. Finally, we prove that the norm of x is given by the Pythagorean formula. We have llxll2 = (x, x) = (s + &, s + sl) = (s, s) + (sl, s'->,the remaining terms being zero since s and sL are orthogonal. This proves (1.18). DEFINITION. Let S be a Jnite-dimensional subspace of a Euclidean space V, and let{e,, . . . , e,} be an orthonormal basis for S. If x E V, the element s dejned by the equation s = 2 (x, ei)ei i=lis called the projection of x on the subspace S. We prove next that the projection of x on S is the solution to the approximation problemstated at the beginning of this section.1.16 Best approximation of elements ,in a Euclidean space by elements in a finite- dimensional subspace THEOREM 1.16. APPROXIMATION THEOREM. Let S be a ,finite-dimensional subspace ofa Euclidean space V, and let x be any element o f V. Then the projection o f x on S is nearer tox than any other element o f S. That is, [f s is the projection o f x on S, we have llx - $11 I IIX - tllfor all t in S; the equality sign holds if and only if t = s. Proof. By Theorem 1.15 we can write x = s + sL, where s E S and s1 E SL. Then,for any t in S, we have x - t = (x - s) + (s - t) .Since s - t E S and x - s = s-l E SL, this is an orthogonal decomposition of x - t, soits norm is given by the Pythagorean formula lb - tl12 = IIX - sly + l/s - tll2.But IIs - tlj2 2 0, so we have IIx - tl12 2 [Ix - sl12, with equality holding if and only ifs = t. This completes the proof.
2 LINEAR TRANSFORMATIONS AND MATRICES2.1 Linear transformations One of the ultimate goals of analysis is a comprehensive study of functions whosedomains and ranges are subsets of linear spaces. Such functions are called transformations,mappings, or operators. This chapter treats the simplest examples, called linear transforma-tions, which occur in all branches of mathematics. Properties of more general transforma-tions are often obtained by approximating them by linear transformations. First we introduce some notation and terminology concerning arbitrary functions. Let V and W be two sets. The symbol T:V+Wwill be used to indicate that T is a function whose domain is V and whose values are in W.For each x in V, the element T(x) in W is called the image of x under T, and we say that Tmaps x onto T(x). If A is any subset of V, the set of all images T(x) for x in A is called theimage of A under T and is denoted by T(A). The image of the domain V, T(V), is the rangeof T. Now we assume that V and Ware linear spaces having the same set of scalars, and wedefine a linear transformation as follows. DEFINITION. If V and Ware linear spaces, a function T: V + W is called a linear trans-formation of V into W if it has the.following two properties: (a) T(x + y) = T(x) + T(y) for all x and y in V, (b) T(cx) = CT(X) for all x in V and all scalars c.These properties are verbalized by saying that T preserves addition and multiplication byscalars. The two properties can be combined into one formula which states that T(ax + by) = aT(x) + bTQ)for all x,y in V and all scalars a and 6. By induction, we also have the more general relationfor any n elements x1, . . . , x,inVandanynscalarsa,,...,a,. 31
32 Linear transformations and matrices The reader can easily verify that the following examples are linear transformations. EXAMPLE 1. The identity transformation. The transformation T: V + V, where T(x) = xfor each x in V, is called the identity transformation and is denoted by Z or by IV. EXAMPLE 2. The zero transformation. The transformation T: V--f V which maps eachelement of V onto 0 is called the zero transformation and is denoted by 0. EXAMPLE 3. Multiplication by ajxed scalar c. Here we have T: V + V, where T(x) = cxfor all x in V. When c = 1 , this is the identity transformation. When c = 0, it is the zerotransformation. EXAMPLE 4. Linear equations. Let V = V,, and W = V, . Given mn real numbers aiL,wherei= 1,2,...,mandk= 1,2,...,n,defineT: V,+V,asfollows: T m a p s e a c hvector x = (x1, . . . , x,) in V, onto the vector y = (,vl, . . . , ym) in V, according to theequations Yi = l$ aikXk for i=l,2,..., m. k=l EXAMPLE 5. Inner product with afixed element. Let V be a real Euclidean space. For afixed element z in V, define T: V -+ R as follows: If x E V, then T(x) = (x, z), the innerproduct of x with z. EXAMPLE 6. Projection on a subspace. Let V be a Euclidean space and let S be a finite-dimensional subspace of V. Define T: V + S as follows: If x E V, then T(x) is theprojection of x on S. EXAMPLE 7. The dzferentiation operator. Let V be the linear space of all real functionsf differentiable on an open interval (a, b). The linear transformation which maps eachfunctionfin V onto its derivativef' is called the differentiation operator and is denoted byD. Thus, we have D: V + W, where D (f) = f' for each f in V. The space W consists ofall derivatives f'. EXAMPLE 8. The integration operator. Let V be the linear space of all real functionscontinuous on an interval [a, b]. IffE V, define g = T(f) to be that function in V given by g(x) = JaZfW dt i f a<x<b.This transformation T is called the integration operator.2.2 Null space and range In this section, Tdenotes a linear transformation of a linear space V into a linear space W. THEOREM 2.1. The set T(V) (the range of T) is a subspace of W. Moreover, T mapsthe zero element of V onto the zero element of W.
Null space and range 33 Proof. To prove that r( I') is a subspace of W, we need only verify the closure axioms.Take any two elements of T(V), say T(x) and r(y). Then T(X) + 3-(y) = T(x + y) , soT(x) + T(y) is in r(V). Also, for any scalar c we have CT(X) = T(cx) , so CT(X) is in T(V).Therefore, T( I') is a subspace of W. Taking c = 0 in the relation T(cx) = CT(X), we findthat T(0) = 0. DEFINITION. The set of all elements in V that T maps onto 0 is called the null space ofT and is denoted by N(T). Thus, we have N(T) = {x 1x E V and T(x) = 0} .The null space is sometimes called the kernel of T. THEOREM 2.2. The null space of T is a subspace of V. Proof. If x and y are in N(T), then so are x + y and cx for all scalars c, since T(x + y) = T(x) + T(y) = 0 and T(cx) = CT(~) = 0. The following examples describe the null spaces of the linear transformations given inSection 2.1. EXAMPLE 1. Identity transformation. The null space is {0}, the subspace consisting ofthe zero element alone. EXAMPLE 2. Zero transformation, Since every element of V is mapped onto zero, thenull space is V itself. EXAMPLE 3. Multiplication by a$xed scalar c. If c # 0, the null space contains only 0.If c = 0, the null space is V. EXAMPLE 4. Linear equations. The null space consists of all vectors (xi, . . . , x,) in V,for which for i=l,2 ,..., m. EXAMPLE 5. Inner product with ajxed element z. The null space consists of all elementsin V orthogonal to z. EXA M P L E 6. Projection on a subspace S. If x E V, we have the unique orthogonaldecomposition x = s + sL (by Theorem I .15). ,'mce T(x) = s, we have T(x) = 0if and only if x = sL . Therefore, the null space is z l, the orthogonal complement of S. EXAMPLE 7. DifSerentiation operator. The null space consists of all functions that areconstant on the given interval. EXAMPLE 8. Integration operator. The null space contains only the zero function.
34 Linear transformations and matrices2.3 Nullity and rank Again in this section T denotes a linear transformation of a linear space V into a linearspace W. We are interested in the relation between the dimensionality of V, of the nullspace N(T), and of the range T(V). If V is finite-dimensional, then the null space is alsofinite-dimensional since it is a subspace of I'. The dimension of N(T) is called the nullityof T. In the next theorem, we prove that the range is also finite-dimensional; its dimensionis called the rank of T. THEOREM 2.3. NULLITY PLUS RANK THEOREM. If V is finite-dimensional, then T(V) isalso finite-dimensional, and we have(2.1) dim N( T> + dim T(V) = dim V .In other words, the nullity plus the rank of a linear transformation is equal to the dimensionof its domain. Proof. Let n = dim V and let e, , . . . , e,beabasisforN(T),wherek = dimN(T)< n.By Theorem 1.7, these elements are part of some basis for V, say the basis(2.2) e,, . . . , ek9 ekflp . . . ? ek+7ywhere k + r = n . We shall prove that the r elements(2.3) T(e,+A . . . 7 T(%,-r)form a basis for T(V), thus proving that dim T(V) = r . Since k + r = n , this also proves(2.1). First we show that the r elements in (2.3) span T(V). If y E T(V), we have y = T(x)for some x in V, and we can write x = clel + * * * + ck+,.ek+,. . Hence, we havesince T(e,) = * *'* = T(e,) = 0. This shows that the elements in (2.3) span T(V). Now we show that these elements are independent. Suppose that there are scalarsck+19 * * - 3 Ck+T such that k+r 2 ciT(ei) = 0 I i=k+lThis implies thatSO the element X = Ck+lek+l + ' ' ' + ck+&k+r is in the null space N(T). This means there
36 Linear transformations and matrices26. Let V be the linear space of all real functions continuous on [a, b]. Iffg V, g = T(f) means that g(x) =/If(t) sin (x - t) dt for a < x 5 b.27. Let V be the space of all real functions twice differentiable on an open interval (a, b). If y E V, define 7"(y) = y" + Py' + Qy , where P and Q are fixed constants.28. Let V be the linear space of all real convergent sequences {x,}. Define a transformation T: V--t V as follows: If x = {x,} is a convergent sequence with limit a, let T(x) = {y,}, where yn = a - x, for n 2 1 . Prove that Tis linear and describe the null space and range of T.29. Let V denote the linear space of all real functions continuous on the interval [-n, ~1. Let S be that subset of Vconsisting of all f satisfying the three equations J:gf(t)costdt =0, j:rf(t)sintdt =O. (a) Prove that S is a subspace of V. (b) Prove that S contains the functions f (x) = cos nx and f (x) = sin nx for each n = 2,3, . . . . (c) Prove that S is infinite-dimensional. Let T: V-t V be the linear transformation defined as follows: Iff E V,g = T(f) means that g(x) = j;, (1 + cos (x - t)}f (t) dt . (d) Prove that T(V), the range of T, is finite-dimensional and find a basis for T(V). (e) Determine the null space of T. (f) Find all real c # 0 and all nonzero f in V such that T(f) = cJ (Note that such an f lies in the range of 7'.)30. Let T: V+ W be a linear transformation of a linear space V into a linear space W. If V is infinite-dimensional, prove that at least one of T(V) or N(T) is infinite-dimensional. [Hint: Assume dim N(T) = k , dim T(V) = r , let e, , . . , ek be a basis for N(T) and let e,, . . . , ek, e,+r, . . . , ek+n be independent elements in V, where n > r. The elements T(e,+,), . . . , T(erc+,) are dependent since n > r. Use this fact to obtain a contradiction.]2.5 Algebraic operations on linear transformations Functions whose values lie in a given linear space W can be added to each other and canbe multiplied by the scalars in W according to the following definition. Let S: V+ W and T: V + W be two functions with a common domain V DEFINITION.and with values in a linear space W. If c is any scalar in W, we define the sum S + T and theproduct CT by the equations (2.4) (S + T)(x) = S(x) + T(x) 3 (CT)(X) = CT(X)for all x in V.
Algebraic operations on linear transformations 37 We are especially interested in the case where V is also a linear space having the samescalars as W. In this case we denote by .Y( V, W) the set of all linear transformations of Vinto W. If S an'd Tare two linear transformations in =.Y( V, W), it is an easy exercise to verify thatS + T and CT are also linear transformations in LZ'(V, W). More than this is true. Withthe opera.tions just defined, the set L?(V, W) itself becomes a new linear space. The zerotransformation serves as the zero element of this space, and the transformation (-1)Tis the neg,ative of T. It is a straightforward matter to verify that all ten axioms for a linearspace are satisfied. Therefore, we have the following. THEOREM 2.4. The set Z'(V, W) of all linear transformations of V into W is a linearspace with the operations of addition and multiplication by scalars de$ned as in (2.4). A more interesting algebraic operation on linear ,transformations is composition ormultiplication of transformations. This operation makes no use of the algebraic structureof a linear space and can be defined quite generally as follows. F IGURE 2.1 Illustrating the composition of two transformations. DEFINI TION . Let U, V, W be sets. Let T: II -+ V be a function with domain U andvalues in V, a.nd let S: V--f W be another function with domain V and values in W. Thenthe composition ST is the function ST: U---f W defined b-y the equation (ST)(x) = S[T(x)] for every x in U. Thus, to map x by the composition ST, we first map x by T and then map T(x) by S.This is illustrated in Figure 2.1. Compo8sition of real-valued functions has been encountered repeatedly in our study ofcalculus, and we have seen that the operation is, in general, not commutative. However,as in the (case of real-valued functions, composition does satisfy an associative law. THEOREM 2.5. If T: u -+ V , S: V + W, and R: W +- X are three functions, then we have R(ST) = (RS)T.
38 Linear transformations and matrices Proof. Both functions R(ST) and (RS)T have domain U and values in X. For each xin U, we haveDWT)ICd = R[W"Ml = R[WWll and WS'X4 = W)LWl = RLW(x)ll,which proves that R(ST) = (RS)T. DEFINITION. Let T: V-t V be a function which maps V into itself. We dejne integralpowers of T inductively as follows: To= I , T" = TT"-1 f o r n>l. Here I is the identity transformation. The reader may verify that the associative lawimplies the law of exponents T"T" = T m+n for all nonnegative integers m and n. The next theorem shows that the composition of linear transformations is again linear. THEOREM 2.6. If U, V, W are linear spaces with the same scalars, and tf T: U -+ Vand S: V -+ W are linear transformations, then the composition ST: U + W is linear. Proof. For all x, y in U and all scalars a and b, we have (ST)(ax + by) = S[T(ax + by)] = S[aT(x) + bT(y)] = aST(x) + bST(y) . Composition can be combined with the algebraic operations of addition and multiplica-tion of scalars in 9(V, W) to give us the following. THEOREM 2.7. Let U, V, W be linear spaces with the same scalars, assume S and T arein Z(V, W), and let c be any scalar. (a) For any function R with values in V, we have (S + T)R = SR + TR and (cS)R = c(SR) . (b) For any linear transformation R: W + U, we have R ( S + T)=RS+RT and R(cS) = c(RS) . The proof is a straightforward application of the definition of composition and is left asan exercise.2.6 Inverses In our study of real-valued functions we learned how to construct new functions byinversion of monotonic functions. Now we wish to extend the process of inversion to amore general class of functions.
Inverses 39 Given a function T, our goal is to find, if possible, another function S whose compositionwith T is the identity transformation. Since composition is in general not commutative,we have to distinguish between ST and TS. Therefore we introduce two kinds of inverseswhich we call left and right inverses. DEFINITION. Given two sets V and Wand a function T: V + W. A function S: T(V) + Vis called a IejYt inverse of T tj'S[T(x)] = x for all x in V, that is, ifwhere r, is the ide,ntity transformation on V. A function R: T(V) + V is called a right inverseof T if T[R(y)] = y for ally in T(V), that is, if TR = IT(V),where I, ( vj is the identity transformation on T(V). EXAMPLE. A function with no left inverse but with two right inverses. Let V = (1, 2)and let W = (0). Define T: V-+ Was follows: T(1) = T(2) = 0. This function has tworight inverses R: W+ V and R' : W---f V given by R(0) = 1, R'(0) = 2.It cannot have a left inverse S since this would require 1 = S[T(l)] = S(0) and 2 = S[T(2)] = S(0).This simple example shows that left inverses need not exist and that right inverses need notbe unique. Every function T: V --f W has at least one right inverse. In fact, each y in T(V) has theform y = T(x) for at least one x in V. If we select one such x and define R(y) = x , thenT[R(y)] = T(x) = y for each y in T(V), so R is a right inverse. Nonuniqueness may occurbecause there may be more than one x in V which maps onto a given y in T(V). We shallprove presently (in Theorem 2.9) that if each y in T(V) is the image of exactly one x in V,then right inverses are unique. First we prove that if a left inverse exists it is unique and, at the same time, is a rightinverse. THEOREM 2.8. A function T: V + W can have at most one left inverse. If T has a leftinverse S, then ,S is also a right inverse. Proof. Assume T has two left inverses, S: T(V)+ Vand S': T(V)+ V. Choose anyy in T(V). We shall prove that S(y) = S'(y) . Now y = T(x) for some x in V, so we have S[T(x)] = x and S' [T(x)] = x ,
40 Linear transformations and matricessince both S and S' are left inverses. Therefore S(y) = x and S'(y) = x , so S(y) = S'@)for all y in T(V). Therefore S = S' which proves that left inverses are unique. Now we prove that every left inverse S is also a right inverse. Choose any element y inr(V). We shall prove that T[S(y)] = y . Since y E 7'( I') , we have y = T(x) for some x in V. But S is a left inverse, so x = S[T(x)] = S(y).Applying T, we get T(x) = T[S@)] . But y = T(x), so y = T[S(y)] , which completes theproof. The next theorem characterizes all functions having left inverses. THEOREM 2.9. A function T: V + W has a left inverse if and only if T maps distinctelements of V onto distinct elements of W; that is, if and only if, for all x and y in V,(2.5) X#Y implies T(x) # T(y). Note: Condition (2.5) is equivalent to the statement (2.6) T(x) = T(y) implies x =y. A function T satisfying (2.5) or (2.6) for all x and y in V is said to be one-to-one on V. Proof. Assume T has a left inverse S, and assume that T(x) = T(y). We wish to provethat x = y . Applying S, we find S[T(x)] = S[T(y)] S'mce S[T(x)] = x and S[T(y)] = y,this implies x = y. This proves that a function with a left inverse is one-to-one on itsdomain. Now we prove the converse. Assume Tis one-to-one on V. We shall exhibit a functionS: T(V) --f V which is a left inverse of T. If y E T(V) , then y = T(x) for some x in V. By (2.6), there is exactly one x in V for which y = T(x). Define S(y) to be this x. That is,we define S on T(V) as follows: S(y) = x means that T(x) = y .Then we have S[T(x)] = x for each x in V, so ST = I,. Therefore, the function S sodefined is a left inverse of T. DEFINITION. Let T: V -+ W be one-to-one on V. The unique left inverse of T (which we know is also a right inverse) is denoted by T-l. We say that T is invertible, and we call T-l the inverse of T. The results of this section refer to arbitrary functions. Now we apply these ideas to linear transformations.
One-to-one linear transformations 412.7 One-to-one linear transformations In this section, V and W denote linear spaces with the same scalars, and T: V-t Wdenotes a linear transformation in Z?( V, W). The linearity of T enables us to express theone-to-one property in several equivalent forms. THEOREM 2.10~. Let T: V -+ W be a linear transformation in Z( V, W). Then the followingstatements are equivalent. (a) T is one-to-one on V. (b) T is invertible and its inverse T-l : T(V) + V is linear. (c) For all x in V, T(x) = 0 implies x = 0. That is, the null space N(T) contains only the zero element of V. Proof. We shall prove that (a) implies (b), (b) implies (c), and (c) implies (a). Firstassume (a) holds. Then T has an inverse (by Theorem 2.'9), and we must show that T-lis linear. Take any two elements u and v in T(V). Then u = T(x) and v = T(y) for somex and y in V. For any scalars a and b, we have a* •k bv = aT(x) + bT(y) = T(ax + by),since T is linear. Hence, applying T-l, we have T-Yau + bv) = ax + by = aT-l(u) + bT-l(v),so T-l is linear.. Therefore (a) implies (b). Next assume that (b) holds. Take any x in V for which T(x) = 0. Applying T-l, wefind that x = T-l(O) = 0, since T-l is linear. Therefore, (b) implies (c). Finally, assu.me (c) holds. Take any two elements u and v in V with T(u) = T(v). Bylinearity, we have T(u - v) = T(u) - T(v) = 0, so u - v = 0. Therefore, Tis one-to-oneon V, and the proof of the theorem is complete. When V is finite-dimensional, the one-to-one property can be formulated in terms ofindependence and dimensionality, as indicated by the next theorem. THEOREM 2.1 I. Let T: V + W be a linear transformatton in 3( V, W) and assume thatV is jinite-dime,nsional, say dim V = n . Then the following statements are equivalent. (a) T is one-to-one on V. (b)Zfe,,..., e, are independent elements in V, then T(e,), . . . , T(e,) are independent elements in T(V). (c) dim T(V) = n . (4 lfh,..., e,} is a basis for V, then {T(e,), . . . , T(e,)} is a basis for T(V). Proof. We shall prove that (a) implies (b), (b) implies (c), (c) implies (d), and (d) implies(a). Assume (a) holds. Let e,, . . . , e, be independent elements of V and consider the
44 Linear transformations and matrices and t(x), respectively, where 44 = p(O) , t(x) = i CkX*+l. k=l Rx0 (a) Let p(x) = 2 + 3x - x2 + x3 and determine the image of p under each of the following transformations: R, S, T, ST, TS, (TS)2, T2S2, S2T2, TRS, RST. (b) Prove that R, S, and Tare linear and determine the null space and range of each. (c) Prove that T is one-to-one on V and determine its inverse. (d) If n 2 1, express ( TS)n and SnTn in terms of I and R.32. Refer to Exercise 28 of Section 2.4. Determine whether T is one-to-one on V. If it is, describe its inverse.2.9 Linear transformations with prescribed values If V is finite-dimensional, we can always construct a linear transformation T: V-+ Wwith prescribed values at the basis elements of V, as described in the next theorem. THEOREM 2.12. Let e,,..., e, be a basis for an n-dimensional linear space V. LetUl,... 7 u, be n arbitrary elements in a linear space W. Then there is one and only one lineartransformation T: V + W such that(2.7) T(%> = u/c for k=l,2 ,..., n.This T maps an arbitrary element x in V as follows:(2.8) I f x =ix,e,, then T(x) = i xkuk. k=l k=l Proof. Every x in V can be expressed uniquely as a linear combination of e,, . . . , e, ,the multipliers x1, . . . , x, being the components of x relative to the ordered basis(e,, . . . , e,). If we define T by (2.8), it is a straightforward matter to verify that T islinear. If x = ek for some k, then all components of x are 0 except the kth, which is 1, so (2.8) gives T(e,) = uk, are required. To prove that there is only one linear transformation satisfying (2.7), let T' be anotherand compute T'(x). We find that =k$IxkT'(eb) = i xkuk = T(x). k=lSince T'(x) = T(x) for all x in V, we have T' = T, which completes the proof. EXAMPLE. Determine the linear transformation T: V, -+ V, which maps the basis elementsi = (1,O) and j = (0, 1) as follows: T(i) = i +i, T(j) = 2i -j.
Matrix representations of linear transformations 45-__- Solution. Nf x == x,i + xz j is an arbitrary element of V,, then T(x) is given by T(x) = x,T(i) + xzT(j) = Xl(i +j> + x,(2i - j) = (x, + 2x& + (Xl - x2)j.2.10 Matrix representations of linear transformations Theorem 2.12 shows that a linear transformation T: V-+ W of a finite-dimensionallinear space V is completely determined by its action on a given set of basis elementse,, . . . , e,. Now, suppose the space W is also finite-dimensional, say dim W = m, and let6, * * *, w, be a basis for W. (The dimensions n and m may or may not be equal.) Since Thas values in IV, each element T(e,) can be expressed uniquely as a linear combination of thebasis elements wl, . . . , w,, say T(e,) = 2 tikwi, i=lwhere t,, , . . . , t,, are the components of T(e,) relative to the: ordered basis (w,, . . . , w,).We shall display the m-tuple (t,, , , . . , tmk) vertically, as follows: hk(2.9) t, tThis array is called a column vector or a column matrix. We have such a column vector foreach of the n elements T(e,), . . . , T(e,). We place them side by side and enclose them in 1 :-one pair of brackets to obtain the following rectangular array: -t11 *-* t1, t21 t22 - * * 2n . . . . . . t ml tnl2 * * * trim-This array is ca.lled a matrix consisting of m rows and n columns. We call it an m by n matrix,or an m x n matrix. The first row is the 1 x n matrix (tl-, , t,,, . . . , tl,). The m x 1matrix displayed in (2.9) is the kth column. The scalars tik are indexed so the first subscripti indicates the row:, and the second subscript k indicates the column in which tik occurs.We call tik the ik-entry or the ik-element of the matrix. The more compact notationis also used to denote the matrix whose ik-entry is tik .
46 Linear transformations and matrices Thus, every linear transformation T of an n-dimensional space V into an m-dimensionalspace W gives rise to an m x n matrix (tik) whose columns consist of the components ofT(e,>, . . . , T(e,) relative to the basis (wl, . . . , w,). We call this the matrix representationof T relative to the given choice of ordered bases (e,, . . . , e,) for Vand (wl, . . . , w,) forW. Once we know the matrix (tik), the components of any element T(x) relative to thebasis (w,, . . . , w,) can be determined as described in the next theorem. THEOREM 2.13. Let T be a linear transformation in 9( V, W), bchere dim V = n anddim W = m. Let (e,, . . . , e,,) and (wl, . . . , w,) be OrderedbasesJor Vand W, respectively,and let (tik) be the m x n matrix whose entries are determined by the equations(2.10) T(e,) = 2 tikWi 3 for k=1,2 ,.,., n. i=lThen an arbitrary element n(2.11) x = zxkek k=lin V with components (x1, . . . , x,) relative to (e,, . , . , e,) is mapped by T onto the element(2.12) T(x) = 2 yiwi i=lin W with components (yl, . . . , y,) relative to (w, , . . . , w,). The yi are related to thecomponents of x by the linear equations(2.13) Y, = i tikXk for i=l,2,.,., m. k=l Proof. Applying T to each member of (2.11) and using (2.10), we obtainwhere each yi is given by (2.13). This completes the proof. Having chosen a pair of bases (e,, . . . , e,) and (+, . . . , MI,) for V and W, respectively,every linear transformation T : V + W has a matrix representation (tik). Conversely, ifwe start with any ~FIII scalars arranged as a rectangular matrix (rik) and choose a pair ofordered bases for V and W, then it is easy to prove that there is exactly one linear trans-formation T: V-t W having this matrix representation. We simply define Tat the basiselements of V by the equations in (2.10). Then, by Theorem 2.12, there is one and onlyone linear transformation T: V + W with these prescribed values. The image T(x) of anarbitrary point x in V is then given by Equations (2.12) and (2.13).
Matrix representations of linear transformations 47_ _ _ _ _ _ _ EXAMPLE. 1. Construction of a linear transformation from a given matrix. Suppose westart with the 2 x 3 matrix r3 1 -21Choose the usual bases of unit coordinate vectors for V, and V,. Then the given matrixrepresents a linear transformation T: V, ---f V, which maps an arbitrary vector (xi, x2, x8)in V3 onto the vector Q1, yZ) in V, according to the linear equations y, = 3x, + xg - 2X,) yz = Xl + o x , + 4x,. EXAMPLE 2. (?OnStrUCtiOn of a matrix representation ?f a giVt?a linear transformation.Let V be the linear space of all real polynomials p(x) of degree < 3. This space has dimen-sion 4, and we choose the basis (I, x, x2, x3). Let D be the differentiation operator whichmaps each polynomial p(x) in V onto its derivative p'(x). We can regard D as a lineartransformation of V into W, where W is the 3-dimensional space of all real polynomialsof degree <: 2'. In W we choose the basis (1, x, x2). To find the matrix representation of Drelative to this (choice of bases, we transform (differentiate) each basis element of V andexpress it as a linear combination of the basis elements of W. Thus, we find that D(1) = 0 = 0 + ox + 0x2, D(x)=1 =1+0x+0x2, D(x2) = 2 x = 0 + 2 X + 0X2, D(x3)=3x2=0+0x+3X2.The coefficients of these polynomials determine the columns of the matrix representation ofD. Therefore, the required representation is given by the following 3 x 4 matrix: i I 0 1 0 0 0 0 2 0 . 0 0 0 3 To emphasize that the matrix representation depends not only on the basis elements butalso on their order, let us reverse the order of the basis elements in Wand use, instead, theordered basis ($3, x, 1). Then the basis elements of V are transformed into the same poly-nomials obtained above, but the components of these polynomials relative to the newbasis (x2, x., 1) appear in reversed order. Therefore, the matrix representation of D nowbecomes [ I 0 0 0 3 0 0 2 0 . 0 1 0 0
48 Linear transformations and matrices Let us compute a third matrix representation for D, usingthebasis (1, 1 + x , 1 + x + x2,1 + x + x2 + x3) for V, and the basis (1, x, x2) for W. The basis elements of Vare trans-formed as follows: D(1) = 0, D(1 + x) = 1, D ( l + x + x2 ) = 1 + 2 x , 2 D ( l +x+x2+x3)=1 +2x+3x ,so the matrix representation in this case is [1 0 1 1 1 0 0 2 2 . 0 0 0 32.11 Construction of a matrix representation in diagonal form Since it is possible to obtain different matrix representations of a given linear transforma-tion by different choices of bases, it is natural to try to choose the bases so that the resultingmatrix will have a particularly simple form. The next theorem shows that we can makeall the entries 0 except possibIy along the diagonal starting from the upper left-hand cornerof the matrix. Along this diagonal there will be a string of ones followed by zeros, thenumber of ones being equal to the rank of the transformation. A matrix (tik) with allentries ti, = 0 when i # k is said to be a diagonal matrix. THEOREM 2.14. Let V and W be finite-dimensional linear spaces, with dim V = n anddim W = m . Assume T E Z( V, W) and let r = dim T(V) denote the rank of T. Then thereexists a basis (e, , . . . , e,) for V and a basis (wl, . . . , w,)for W such that(2.14) T(eJ = wi for i=1,2 ,..., r,and(2.15) T(e,)=O f o r i=r+l,...,n.Therefore, the matrix (tir) of T relative to these bases has all entries zero except for the rdiagonal entries tll = tzz = ' * ' = t,, = 1 . Proof. First we construct a basis for W. Since T(V) is a subspace of W with dim T(V) =r, the space T(V) has a basis of r elements in W, say wr, . . . , w,, By Theorem 1.7, theseelements form a subset of some basis for W. Therefore we can adjoin elements w,+~, . . . ,w, so that(2.16) (Wl,. * * 7 WT, Wr+l, * . . 2 w,)is a basis for W.
Construction of a matrix representation in diagonal form 49 Now we construct a basis for V. Each of the first r elements u'~ in (2.16) is the image of atleast one element in V. Choose one such element in V and call it e, . Then T(e,) = wji fori= 1,2,..., r so (2.14) is satisfied. Now let k be the dimension of the null space N(T).By Theorem 2.3 we have n = k + r. Since dim N(T) =: k, the space N(T) has a basisconsisting of k elements in V which we designate as e,,, , . . . , er.tlc. For each of theseelements, Equation (2.15) is satisfied. Therefore, to comlplete the proof, we must showthat the ordered set(2.17) (e,, . . . , e,, ql, . . . , q.+k)is a basis flar V. Since dim V = n = r + k, we need only show that these elements areindependent. Suppose that some linear combination of th,em is zero, say(2.18)Applying 7' and using Equations (2.14) and (2.19, we find thatBut wl,..., w, are independent, and hence c1 = * * * = c, = 0. Therefore, the first rterms in (2.18) a.re zero, so (2.18) reduces to r+k i&lc,ei = O.But e,,, , * * * , are independent since they form a basis for N(T), and hence c,+~ = er+k. ..=c r+k = 0. Therefore, all the ci in (2.18) are zero, S,D the elements in (2.17) form abasis for V. This completes the proof. EXAMPLE. We refer to Example 2 of Section 2.10, where D is the differentiation operatorwhich maps the space V of polynomials of degree < 3 into the space W of polynomials ofdegree 52. In this example, the range T(V) = W, so T has rank 3. Applying the methodused to prove Theorem 2.14, we choose any basis for W, for example the basis (1, x, x2).A set of polynomials in V which map onto these elements is given by (x, +x2, +x3). Weextend this set to get a basis for V by adjoining the constant polynomial 1, which is a basisfor the null space of D. Therefore, if we use the basis (x, Zx 2, &x3, 1) for V and the basis l(1, x, x2) for W, the corresponding matrix representation for D has the diagonal form 1 1 1 0 0 0 0 1 0 0 . 10 0 1 01
Linear spaces of matrices 51 (c) Use the basis (q , e,) for V and find a new basis of the form (el + uez, 2e, + be,) for W, relative to which the matrix of Twill be in diagonal form. In the linear space of all real-valued functions, each of the following sets is independent andspans a finite-dimensional subspace V. Use the given set as a basis for V and let D: VA V bethe differentiation operator. In each case, find the matrix of D and of D2 relative to this choiceof basis.11. (sin x, cos x). 15. (-cos x, sin x).12. (1, x, e5). 16. (sin x, cos x., x sin x, x cos x).13. (1, I -t x, I + x + er). 17. (eX sin x, e2 cos x).14. (e", xe"). 18. (ezz sin 3x, ezs cos 3x).19. Choose the basis (1, x, x2, x3) in the linear space V of all real polynomials of degree 13. Let D denote the differentiation operator and let T: V + 1' be the linear transformation which map'sp(x:, onto x$(x). Relative to the given basis, determine the matrix of each of the followingtransformations: (a) T; (b) DT; (c) TD; (d) TD - DT; (e) T2; (f) T2D2 - D2T2.20. Refer to Exercise 19. Let W be the image of Vunder TD. Find bases for Vand for W relative to which the matrix of TD is in diagonal form.2.13 Linear spaces of matrices We have seen how matrices arise in a natural way as representations of linear trans-formations. Matrices can also be considered as objects existing in their own right, withoutnecessarily being connected to linear transformations. As such, they form another class ofmathematical objects on which algebraic operations can be defined. The connectionwith linear transformations serves as motivation for these definitions, but this connectionwill be ignored for the moment. Let nz and n be two positive integers, and let I,,, be the set of all pairs of integers (i,j)such that 1 I; i 5 m, 1 <j 5 PI. Any function A whose domain is I,,, is called an m x nmatrix. The flJnction value A(i, j) is called the g-entry or ij-element of the matrix and willbe denoted also by aij . It is customary to display all the function values in a rectangulararray consisting of m rows and n columns, as follows: azl az2 **a azn . .The elements a;j may be arbitrary objects of any kind. Usually they will be real or complexnumbers, but sometimes it is convenient to consider matrices whose elements are otherobjects, for example, functions. We also denote matrices by the more compact notation A = (u,,)~;~~ or A = (aij).If m = n , the matr.ix is said to be a square matrix. A 1 x n matrix is called a row matrix;an m x 1 matrix is called a column matrix.
52 Linear transformations and matrices Two functions are equal if and only if they have the same domain and take the samefunction value at each element in the domain. Since matrices are functions, two matricesA = (a,J and B = (bJ are equal if and only if they have the same number of rows, thesame number of columns, and equal entries aij = bij for each pair (i,j). Now we assume the entries are numbers (real or complex) and we define addition ofmatrices and multiplication by scalars by the same method used for any real- or complex-valued functions. DEFINITION. If A = (aij) and B = (bij) are two m x n matrices and if c is any scalar,we define matrices A + B and CA as follows: A + B = (a,j + b,j>, CA = (caij).The sum is defined only when A and B have the same size. EXAMPLE. If 1 2 - 3 A = -1 0 4I and B = [: -1 i],then we haveA+.=[; -; -:]. 2A= [-'z ; -;], (-1)B= [I; ; I:]. We define the zero matrix 0 to be the m x n matrix all of whose elements are 0. -Withthese definitions, it is a straightforward exercise to verify that the collection of all m x nmatrices is a linear space. We denote this linear space by M,,,. If the entries are realnumbers, the space M,,, is a real linear space. If the entries are complex, M,,, is a complexlinear space. It is also easy to prove that this space has dimension mn. In fact, a basis forM consists of the mn matrices having one entry equal to 1 and all others equal to 0.Fo?"example, the six matricesform a basis for the set of all 2 x 3 matrices.2.14 Isomorphism between linear transformations and matrices We return now to the connection between matrices and linear transformations. Let Vand W be finite-dimensional linear spaces with dim V = n and dim W = m. Choose abasis (e,, . . . , e,) for V and a basis (wl, . . . , w,) for W. In this discussion, these bases arekept fixed. Let p(V, W) denote the linear space of all linear transformations of V into W. If T E P(V, W), let m(T) denote the matrix of T relative to the given bases. We recallthat m(T) is defined as follows.
Zsomorphism between linear transformations and matrices 53 The image of each basis element e, is expressed as a linear combination of the basiselements in IV:(2.19) T(e,) = g tikWi for k=l,2 ,..., n. i=lThe scalar multipliers ci, are the ik-entries of m(T). Thus, we have(2.20) 47 = (4,>~&. Equation (2.20) defines a new function m whose domain is L?'(V, W) and whose valuesare matrices in M,, ~. Since every m x n matrix is the,matrix m(T) for some Tin A?( V, W),the range of m is ii,,,. The next theorem shows that the transformation m: LZ(V, W) +M 112.11 is linear and one-to-one on LY( V, W). THEOREM 2.15. ISOMORPHISM THEOREM. For all S and 1" in P(V, W) and all scalarsc, we have 4s + T> = m(S) + m(T) and m(cT) = cm(T).Moreover, m(S) = m(T) implies S = T,so m is one-to-one on 9(V, W). Proof. The matrix m(T) is formed from the multipliers ti, in (2.19). Similarly, thematrix m(S) is formed from the multipliers sik in the equations(2.21) S(e,) = 5 SikWi for k=1,2 ,..., n. i=lSince we have andwe obtainm(S + 73 = (sik + tile) = m(S) + m(T) and m(cT) = (ctJ = cm(T). This provesthat m is linear. To prove that m is one-to-one, suppose that m(S) = m(Z3, where S = (sik) and T = (t&. Equations (2.19) and (2.21) show that S(e,) = T(e,) for each basis element e,, so S(x) = T(x) for all x in V, and hence S = T. Note: The function m is called an isomorphism. For a given choice of bases, m establishes a one-to-one correspondence between the set of linear transformations U(V, W) and the set of m x n matrices M,,, . The operations of addition and multipli- cation by scalars are preserved under this correspondence. The linear spaces -Y(V, W) and Mm,, are said to be isomorphic. Incidentally, Theorem 2.11 shows that the domain of a one-to-one linear transformation has the same dimension as its range. Therefore, dim Y(V, IV) = dim M,,, = mn . If V = Wand if we choose the same basis in both V and W, then the matrix m(Z) whichcorresponds to the identity transformation I: V ---f V is an n x n diagonal matrix with each
54 Linear transformations and matricesdiagonal entry equal to 1 and all others equal to 0. This is called the identity or unit matrixand is denoted by I or by I,.2.15 Multiplication of matrices Some linear transformations can be multiplied by means of composition. Now we shalldefine multiplication of matrices in such a way that the product of two matrices correspondsto the composition of the linear transformations they represent. We recall that if T: U--f V and S: V + Ware linear transformations, their compositionST: U--j W is a linear transformation given by ST(x) = S[T(x)] f o r a l l x i n U.Suppose that U, V, and Ware finite-dimensional, say dim U = n, dim V=p, dim W= m.Choose bases for U, V, and W. Relative to these bases, the matrix m(s) is an m x pmatrix, the matrix T is a p x n matrix, and the matrix of ST is an m x n matrix. Thefollowing definition of matrix multiplication will enable us to deduce the relation m(ST) =m(S)m(T). This extends the isomorphism property to products. DEFINITION. Let A be any m x p matrix, and let B be any p x n matrix, say A = (u,,);;Z)~ and B = (bij);,8jn=1.Then the product AB is defined to be the m x n matrix C = (cJ whose ij-entry is given by(2.22) Note: The product AB is not defined unless the number of columns of A is equal to the number of rows of B. If we write Ai for the ith row of A, and Bi for thejth column of B, and think of these asp-dimensional vectors, then the sum in (2.22) is simply the dot product Ai * Bj. In otherwords, the q-entry of AB is the dot product of the ith row of A with thejth column of B: AB = (Ai . B');3"1.Thus, matrix multiplication can be regarded as a generalization of the dot product. ExAMPLnl. LetA= [-: : i]andB= [ --!I. S i n c e A i s 2 x 3andBis3 x 2 ,
58 Linear transformations and matrices 1 0 9. Let A = Prove that A2 = 2A - Z and compute Aloo. [ -1 1 I *10. Find all 2 x 2 matrices A such that A2 = 0.11. (a) Prove that a 2 x 2 matrix A commutes with every 2 x 2 matrix if and only if A commutes with each of the four matrices (b) Find all such matrices A.12. The equation A 2 = Z is satisfied by each of the 2 x 2 matrices where b and c are arbitrary real numbers. Find all 2 x 2 matrices A such that A2 = I.U.IfA=[-: -t] andB=[i 11, find 2 x 2 matrices C and D such that AC = B andDA=B.14. (a) Verify that the algebraic identities (A + B)2 = A2 + 2AB + B2 and (A + B)(A - B) = A2 - B2 do not hold for the 2 x 2 matrices A = [t -t]andB=[: J. (b) Amend the right-hand members of these identities to obtain formulas valid for all square matrices A and B. (c) For which matrices A and B are the identities valid as stated in (a)?2.17 Systems of linear equations Let A = (aJ be a given m x n matrix of numbers, and let cl, . . . , c, be m furthernumbers. A set of m equations of the form(2.23) $Fixxk = ci for i = 1, 2, . . . , m,is called a system of m linear equations in n unknowns. Here x1, . . . , X, are regarded asunknown. A solution of the system is any n-tuple of numbers (x1, . . . , x,) for which all theequations are satisfied. The matrix A is called the coefJicient-matrix of the system. Linear systems can be studied with the help of linear transformations. Choose the usualbases of unit coordinate vectors in V, and in V,,. The coefficient-matrix A determines a
Systems of linear equations 59linear transformation, T: V, + V,, which maps an arbitrary vector x = (xi, . . . , x,) in V,onto the vector y = (yl, . . . , ym) in V, given by the m linear equations yi = i aikXk for i=1,2 ,..., m. k=lLet c = (cr , . . . , c,) be the vector in V, whose components are the numbers appearing insystem (2.23). This system can be written more simply as; T(x) = c.The system has a solution if and only if c is in the range of T. If exactly one x in V,, mapsonto c, the system has exactly one solution. If more than one x maps onto c, the systemhas more than one solution. EXAMPLE: 1. A system with no solution. The system x + y = 1, x + y = 2 has nosolution. 'The sum of two numbers cannot be both I and 2. EXAMPLE 2. .4 system with exactly one solution. The system x + y = 1, x - y = 0 hasexactly one solution: (x, y) = (4, 4). EXAMPLE: 3. .4 system with more than one solution. The system x + y = 1 , consistingof one equation in two unknowns, has more than one solution. Any two numbers whosesum is 1 gives ii solution. With each linear system (2.23), we can associate another system &kXk = 0 for i = 1, 2, . . . , m,obtained by replacing each ci in (2.23) by 0. This is called the homogeneous system corre-sponding to (2.23). If c Z 0, system (2.23) is called a nonhomogeneous system. A vectorx in V, will satisfy the homogeneous system if and only if T(x) = 0,where T is, the linear transformation determined by the coefficient-matrix. The homogene-ous system always has one solution, namely x = 0, but it may have others. The set ofsolutions of the homogeneous system is the null space of 7: The next theorem describes therelation between solutions of the homogeneous system and those of the nonhomogeneoussystem. THEOREM 2.18. Assume the nonhomogeneous system (2.23) has a solution, say 6. (a) If a vector x is a solution of the nonhomogeneous system, then the vector v = x - b is a solution of the corresponding homogeneous system. (b) If a vector v is a solution of the homogeneous+ system, then the vector x = v + b is a solution #of the nonhomogeneous system.
60 Linear transformations and matrices Proof. Let T: V, + V, be the linear transformation determined by the coefficient-matrix, as described above. Since b is a solution of the nonhomogeneous system we haveT(b) = c . Let x and v be two vectors in V, such that v = x - b. Then we have T(u) = T(x - b) = T(x) - T(b) = T(x) - c.Therefore T(x) = c if and only if T(v) = 0. This proves both (a) and (b). This theorem shows that the problem of finding all solutions of a nonhomogeneoussystem splits naturally into two parts: (1) Finding all solutions v of the homogeneoussystem, that is, determining the null space of T; and (2) finding one particular solution b ofthe nonhomogeneous system. By adding b to each vector v in the null space of T, we therebyobtain all solutions x = v + b of the nonhomogeneous system. Let k denote the dimension of N(T) (the nullity of T). If we can find k independentsolutions vi, . . . , vk of the homogeneous system, they will form a basis for N(T), and wecan obtain every v in N(T) by forming all possible linear combinations u = t,v, + * * * + t,v, ,where t,, . . . , t, are arbitrary scalars. This linear combination is called the general solutionof the homogeneous system. If b is one particular solution of the nonhomogeneous system,then all solutions x are given by x = b + t,V, + * * . + t,V,.This linear combination is called the general solution of the nonhomogeneous system.Theorem 2.18 can now be restated as follows. THEOREM 2.19. Let T: V, -+ V, be the linear transformation such that T(x) = y, wherex = (x1,*--, x,), y = Q1, . . . , ym> and Yi = &ikXk for i=l,2 ,,.., m. k=lLet k denote the nullity of T. If vl, . . . , vk are k independent solutions of the homogeneoussystem T(x) = 0 , and if b is one particular solution of the nonhomogeneous system T(x) = c ,then the general solution of the nonhomogeneous system is x = b + t,v, + . . . + tkvk,where t,, . . . , t, are arbitrary scalars. This theorem does not tell us how to decide if a nonhomogeneous system has a particularsolution b, nor does it tell us how to determine solutions vl, . . . , vk of the homogeneoussystem. It does tell us what to expect when the nonhomogeneous system has a solution.The following example, although very simple, illustrates the theorem.
Computation techniques 61 EXAMPLE. The system x + y = 2 has for its associated homogeneous system the equationx + y = 0. Therefore, the null space consists of all vectors in V, of the form (t, -t),where t is arbitrary. Since (t, -t) = t(1, -I), this is a one-dimensional subspace of V,with basis (1, -- 1). A particular solution of the nonhomogeneous system is (0,2). There-fore the general solution of the nonhomogeneous system is given by (x,y> = (0,2) + t(1, -1) or x = t, y=2-tt,where t is arbitrary.2.18 Computation techniques We turn now to the problem of actually computing the solutions of a nonhomogeneouslinear system. Although many methods have been developed for attacking this problem,all of them require considerable computation if the system is large. For example, to solvea system of ten equations in as many unknowns can require several hours of hand com-putation, (even with the aid of a desk calculator. We shall discuss a widely-used method, known as the Gauss-Jordan elimination method,which is relatively simple and can be easily programmed for high-speed electronic computingmachines. The method consists of applying three basic types of operations on the equationsof a linear system: (1) Interchanging two equations; (2) Mui'tiply,ing all the terms of an equation by a nonzero scalar: (3) Adding tlg one equation a multiple of another.Each time we perform one of these operations on the system we obtain a new system havingexactly the same solutions. Two such systems are called equivalent. By performing these operations over and over again in a systematic fashion we finally arrive at an equivalent system which can be solved by inspection. We shall illustrate the method with some specific examples. It will then be clear how the method is to be applied in general. EXAMPLE 1. .4 system with a unique solution. Consider .the system 2x-5y+42= - 3 x-2y+ z=5 x-4y+6z=lO.This partiizular system has a unique solution, x = 124, y = 75, z = 31 , which we shallobtain by the Gauss-Jordan elimination process. To save labor we do not bother to copythe letters x, y,, z and the equals sign over and over again, but work instead with the aug-mented matrix 2 -5 4 -3(2.24) l-2 1 5 1 -4 6 10
62 Linear transformations and matricesobtained by adjoining the right-hand members of the system to the coefficient matrix. Thethree basic types of operation mentioned above are performed on the rows of the augmentedmatrix and are called row operations. At any stage of the process we can put the lettersx, y, z back again and insert equals signs along the vertical line to obtain equations. Ourultimate goal is to arrive at the augmented matrix [ 1 1 0 0 124(2.25) 0 10 75 0 0 1 3 1after a succession of row operations. The corresponding system of equations is x = 124,y = 75, z = 31 , which gives the desired solution. The first step is to obtain a 1 in the upper left-hand corner of the matrix. We can do thisby interchanging the first row of the given matrix (2.24) with either the second or thirdrow. Or, we can multiply the first row by h. Interchanging the first and second rows, we get ! I l-2 1 5 2 -5 4 -3 . 1 -4 6 10 The next step is to make all the remaining entries in the first column equal to zero, leavingthe first row intact. To do this we multiply the first row by -2 and add the result to thesecond row. Then we multiply the first row by -1 and add the result to the third row.After these two operations, we ob tstin(2.26) 0 '1 - -2 o 2 -1 2 5 1 -13 Now we repeat the process on the smaller matrix [ 1: : / -':I which appears 1 5 5 . iadjacent to the two zeros. We can obtain a 1 in its upper left-hand corner by multiplyingthe second row of (2.26) by - 1 . This gives us the matrix 0 o-2 l-2 1 -2 5 1Multiplying the second row by 2 and adding the result to the third, we get 13 5 5 1 . l - 2 1 5(2.27) [ 0 1 - 2 13 I . 0 0 1 31
Computation techniques 63At this stage, the corresponding system of equations is given by x-2y+ z= 5 y - 22 = 13 2 = 31.These equations can be solved in succession, starting with the third one and workingbackwards, to give usz = 31, y = 13 + 22 = 13 + 62 = 75, x = 5 + 2~ - z = 5 + 150 - 31 = 124.Or, we can continue the Gauss-Jordan process by making all the entries zero above the [ 1diagonal elernents in the second and third columns. MultiFllying the second row of (2.27)by 2 and adding rhe result to the first row, we obtain 001 0 0 1 -3 -2 1 31 31 13 .Finally, we multiply the third row by 3 and add the result to the first row, and then multiplythe third row by :2 and add the result to the second row to get the matrix in (2.25). EXAMPLE 2. A system with more than one solution. Consider the following system of 3equations in 5 unknowns: 2x-55y+4z+ u-vu-3(2.28) x-2y+ z - u+v=5 x-44y+62+2u--v=lO.The corresponding augmented matrix is [ 2-5 1 1 -2 -4 4 6 l-l 2 1 -1 -1 1 -3 10 5. 1 1The coefficients of x, y, z and the right-hand members are the same as those in Example 1.If we perform the same row operations used in Example 1, we: finally arrive at the augmentedmatrix 1 0 0 -16 19 124 0 1 0 -9 11 75 . 0 0 1 -3 4 31
64 Linear transformations and matricesThe corresponding system of equations can be solved for x, y, and z in terms of u and u,giving us x = 124 + 162~ - 190 y = 75+ 9U-110 z = 3 1 + 3u- 4v.If we let u = t, and u = t,, where t, and t2 are arbitrary real numbers, and determinex, y, z by these equations, the vector (x, y, z, U, u) in V, given by (x,y,z, u, u) = (124 + 16t, - 19t,, 75 + 9t, - llt,, 31 + 3t, - 4t,, t,, t,)is a solution. By separating the parts involving t, and t,, we can rewrite this as follows: (x,y,z, u, u) = (124, 75, 31,0,0) + t1(16, 9,3, 1,0) + &C-19, -11, -4,O, 1)sThis equation gives the general solution of the system. The vector (124, 75, 31,0,0) is aparticular solution of the nonhomogeneous system (2.28). The two vectors (16, 9, 3, 1,0)a n d ( - 1 9 , -11, -4 , 0, 1) are solutions of the corresponding homogeneous system. Sincethey are independent, they form a basis for the space of all solutions of the homogeneoussystem. EXAMPLE 3. A system with no solution. Consider the system 2x-5y+4z= - 3(2.29) x-2y+ z = 5 x-4y+5z= 1 0 .This system is almost identical to that of Example 1 except that the coefficient of z in thethird equation has been changed from 6 to 5. The corresponding augmented matrix is i 2 1 l-2 -5 -4 451 -3 10 5. I iApplying the same row operations used in Example 1 to transform (2.24) into (2.27), we 1arrive at the augmented matrix(2.30) 0 0 l-2 0 1 -2 0 1 31135 .When the bottom row is expressed as an equation, it states that 0 = 31. Therefore theoriginal system has no solution since the two systems (2.29) and (2.30) are equivalent.
Inverses of square matrices 65 In each of the foregoing examples, the number of equations did not exceed the numberof unknowns. If there are more equations than unknowm, the Gauss-Jordan process isstill applicable. For example, suppose we consider the syste:m of Example 1, which has thesolution x = 124 , y = 75, z = 31. If we adjoin a new equation to this system which is alsosatisfied by the same triple, for example, the equation 2x - 3~ + z = 54, then theelimination process leads to the agumented matrix 0 10 75 0 0 1 3 1 000 0with a row of zeros along the bottom. But if we adjoin a new equation which is not satisfiedby the triple (124, 75, 31), for example the equation x + y + z = 1, then the eliminationprocess leads to an augmented matrix of the form 1 0 0 124where a # 0. The last row now gives a contradictory equation 0 = a which shows thatthe system has no solution.2.19 Inverses of square matrices Let A = (a,J be a square n x n matrix. If there is another n x n matrix B such thatBA = I, where Z is the n x n identity matrix, then A is called nonsinpdar and B is called aleft inverse of A. Choose the usual basis of unit coordinate vectors in V, and let T: V, + V, be the lineartransformation with matrix m(7') = A. Then we have the following. THEOREM 2.20. The matrix A is nonsingular if and only if T is invertible. If BA = Ithen B = m(T-I). Proof. Assume that A is nonsingular and that BA = I. We shall prove that T(x) = 0implies x = 0. Given x such that T(x) = 0, let X be the n x 1 column matrix formedfrom the components of x. Since T(x) = 0, the matrix product AX is an n x 1 columnmatrix consisting of zeros, so B(AX) is also acolumn matrixof zeros. But B(AX) = (BA)X =IX = X, so every component of x is 0. Therefore, Tis invertible, and the equation TT-l= Zimplies that m(73m(T-l) = Z or Am(T-l) = I. Multiplying on the left by B, we findm(T-l) = B , Conversely, if T is invertible, then T-lT is the identity transformation som(T-l)m(T)i.s the identity matrix. Therefore A is nonsingular and m(F)A = I.
66 Linear transformations and matrices All the properties of invertible linear transformations have their counterparts for non-singular matrices. In particular, left inverses (if they exist) are unique, and every leftinverse is also a right inverse. In other words, if A is nonsingular and BA = I, thenAB = I. We call B the inverse of A and denote it by A-l. The inverse A-' is also non-singular and its inverse is A. Now we show that the problem of actually determining the entries of the inverse of anonsingular matrix is equivalent to solving n separate nonhomogeneous linear systems. Let A = (a,J be nonsingular and let A-l = (hij) be its inverse. The entries of A andA-l are related by the II? equations(2.31)where aij = 1 if i = j , and dij = 0 if i # j. For each fixed choice of j, we can regard thisas a nonhomogeneous system of n linear equations in n unknowns bli, bzj, . . . , b,, . SinceA is nonsingular, each of these systems has a unique solution, the jth column of B. Allthese systems have the same coefficient-matrix A and differ only in their right members.For example, if A is a 3 x 3 matrix, there are 9 equations in (2.31) which can be expressedas 3 separate linear systems having the following augmented matrices :If we apply the Gauss-Jordan process, we arrive at the respective augmented matricesIn actual practice we exploit the fact that all three systems have the same coefficient-matrixand solve all three systems at once by working with the enlarged matrix 1 all 62 al3 1 0 0 a21 az2 a23 0 1 0 . i a3, a32 a33 0 0 1The elimination process then leads to
68 Linear transformations and matrices (b) Determine all solutions of the system 5x +2y -6z+2u= - 1 x - y + z - u = - 2 x+y+z = 6. /11. This exercise tells how to determine all nonsingular 2 x 2 matrices. Prove that [1 a b Deduce that is nonsingular if and only if ad - bc # 0, in which case its inverse is c d Determine the inverse of each of the matrices in Exercises 12 through 16. 1 '1 2 3 4 0 1 2 3 15. 0 0 12' I p 0 0 1_1 -0 1 0 0 0 o-13. 2 0 2 0 0 0 0 3 0 1 0 0 16. 0 0 1 0 2 0 ' 0 0 0 3 0 114. _0 0 0 0 2 0 !2.21 Miscellaneous exercises on matrices 1. If a square matrix has a row of zeros or a column of zeros, prove that it is singular. 2. For each of the following statements about II x n matrices, give a proof or exhibit a counter example. (a) If AB + BA = 0, then A2B3 = B3A2. (b) If A and B are nonsingular, then A + B is nonsingular. (c) If A and B are nonsingular, then AB is nonsingular. (d) If A, B, and A + B are nonsingular, then A - B is nonsingular. (e) If A3 = 0, then A - Z is nonsingular. (f) If the product of k matrices A, . . . A, is nonsingular, then each matrix Ai is nonsingular.
Miscellaneous exercises on mawices 69 find a nonsingular matrix P such that P-'AP = [ I a i4. The matrix A = , where i2 = -1,a =$(l + $$,andb =$(l - &),hastheprop- i b erty that A2 = A. Describe completely all 2 x 2 matrices A with complex entries such that A2=A.5. If A2 = A, prove that (A + I)" = Z f (2k - l)A.6. The special theory of relativity makes use of a set of equations of the form x' = a(x - ut), y' = y, z' = z, t' = a(t - vx/c2). Here v represents the velocity of a moving object, c the speed of light, and a = c/Jc2 - v2, where Ju] < c . The linear transformation which maps the two-dimensional vector (x, t) onto (x', t') is called a L,orentz transformation. Its matrix relative to the usual bases is denoted by L(v) and is given by [ I, 1 -v L(u) = a --DC-Z 1 Note that L(v) is nonsingular and that L(O) = I. Prove that L(v)L(u) = L(w), where w = (u + v)c2/(uv + c2). In other words, the product of two Lorentz transformations is another Lorentz transformation.7. If we interchange the rows and columns of a rectangular matrix A, the new matrix so obtained is called the transpose of A and is denoted by At. For example, if we have [ 1 1 2 3 A = 4 5 6 ' Prove that transposes have the following properties : (a) (At)t = A (b) (A + B)t = At + Bt. (c) (cA)~ = cAt, (d) (AB)t = B"A". (e) (At)-l = (A-l)t if A is nonsingular.8. A square matrix A is called an orthogonal matrix if AA t := I. Verify that the 2 x 2 matrix is orthogonal for each real 0. If A is any n x II orthogonal matrix, prove [::e" z] that its rows, considered as vectors in V, , form an orthonormal set. 9. For each of the following statements about n x n matrices, give a proof or else exhibit a counter example. (a) If A and B are orthogonal, then A + B is orthogonal. (b) If A and B are orthogonal, then AB is orthogonal. (c) If A and B are orthogonal, then B is orthogonal.10. Hadumard matrices, named for Jacques Hadamard (1865-1963), are those n x n matrices with the following properties: I. Each entry is 1 or -1 . II. Each row, considered as a vector in V, , has length &. III. The dot product of any two distinct rows is 0. Hadamard matrices arise in certain problems in geometry and the theory of numbers, and they have been applied recently to the construction of optimum code words in space com- munication. In spite of their apparent simplicity, they present many unsolved problems. The
70 Linear transformations and matrices main unsolved problem at this time is to determine all n for which an n x n Hadamard matrix exists. This exercise outlines a partial solution. (a) Determine all 2 x 2 Hadamard matrices (there are exactly 8). (b) This part of the exercise outlines a simple proof of the following theorem: ZfA is an n x n Hadamard matrix, where n > 2, then n is a multiple of 4. The proof is based on two very simple lemmas concerning vectors in n-space. Prove each of these lemmas and apply them to the rows of Hadamard matrix to prove the theorem. LEMMA 1. If X, Y, Z are orthogonal vectors in V, , then we have (x+ Y)*(X+Z)= lpq2. LEMMA 2. Write X=(x, ,..., xn), Y=(yl ,..., yn), Z=(z, ,..., zn). Zf eachcomponent xi, yi, zi is either 1 or - 1 , then the product (xi + yi)(xi + zi) is either 0 or 4.
3 DETERMINANTS3.1 Introduction In many applications of linear algebra to geometry and analysis the concept of adeterminant plays an important part. This chapter studies the basic properties of determi-nants and some of their applications. Determinants of order two and three were intoduced in Volume I as a useful notation forexpressing certain formulas in a compact form. We recall 'that a determinant of order twowas defined by the formula(3.1) I I Qll Qlz Q21 a22 =Despite similarity in notation, the determinant Q l l Q 2 2 - Ql2Q21 * Qll 'I2 (written with vertical bars) is Ia21 Q22 Iconceptually distinct from the matrix (wn'tt en with square brackets). Thedeterminant is ;i number assigned to the matrix according to Formula (3.1). To emphasizethis connection we also write Determinants of order three were defined in Volume I in terms of second-order determi-nants by the formulaThis chapter treats the more general case, the determinant of a square matrix of order nfor any integer n 2 1 . Our point of view is to treat the: determinant as a function which 71
72 Determinantsassigns to each square matrix A a number called the determinant of A and denoted by det A.It is possible to define this function by an explicit formula generalizing (3.1) and (3.2).This formula is a sum containing n! products of entries of A. For large n the formula isunwieldy and is rarely used in practice. It seems preferable to study determinants fromanother point of view which emphasizes more clearly their essential properties. Theseproperties, which are important in the applications, will be taken as axioms for a determi-nant function. Initially, our program will consist of three parts: (1) To motivate thechoice of axioms. (2) To deduce further properties of determinants from the axioms.(3) To prove that there is one and only one function which satisfies the axioms. a l al2 al33.2 Motivation for the choice of axioms for a determinant function In Volume I we proved that the scalar triple product of three vectors A,, A,, A, in 3-spacecan be expressed as the determinant of a matrix whose rows are the given vectors. Thus [1we have A, x A, - A, = det aZ1 aZ2 aZ3 , a31 a32 a33where Al = (a,, , a12, a13>, A2 = (azl , a22, a23>, and A3 = (a31, a32T a33) - If the rows are linearly independent the scalar triple product is nonzero; the absolutevalue of the product is equal to the volume of the parallelepiped determined by the threevectors A,, A,, A,. If the rows are dependent the scalar triple product is zero. In this casethe vectors A,, A,, A, are coplanar and the parallelepiped degenerates to a plane figure ofzero volume. Some of the properties of the scalar triple product will serve as motivation for the choiceof axioms for a determinant function in the higher-dimensional case. To state theseproperties in a form suitable for generalization, we consider the scalar triple product as afunction of the three row-vectors A,, A,, A,. We denote this function by d; thus, d(A,,A,,A,)=A, X A2*A3.We focus our attention on the following properties: (a) Homogeneity in each row. For example, homogeneity in the first row states that dOA,, A,, A31 = t 44, A2 9 A31 for every scalar t . (b) Additivity in each row. For example, additivity in the second row states that 441, 4 + C, ~43) = 44, A,, A31 + d(A,, C, A31 for every vector C. (c) The scalar triple product is zero if two of the rows are equal. (d) Normalization: d(i,j, k) = 1, w h e r e i=(l,O,O), j=(O,l,O), k=(O,O,l).
A set of axioms for a determinant function 73 Each of these properties can be easily verified from properties of the dot and crossproduct. Some of these properties are suggested by the geometric relation between the scalartriple product and the volume of the parallelepiped determined by the geometric vectorsA,, A,, A:,. The geometric meaning of the additive prc'perty (b) in a special case is ofparticular interest. If we take C = A, in (b) the second term on the right is zero because of (c), and relation (b) becomes(3.3) 44, A, + A,, 4) = d(A, , A,, A&.This property is illustrated in Figure 3.1 which shows a parallelepiped determined byA,, A,, A,, and another parallelepiped determined by A,, A, + AZ, A,. Equation (3.3)merely states that these two parallelepipeds have equal volumes. This is evident geometri-cally because the parallelepipeds have equal altitudes and bases of equal area. Volume = d(A , , A 2r A 3) Volume =: d(A ,, A, + A,, AZ) FI G U R E 3.1 Geometric interpretation of the property d(A,, A,, AS) = d(A,, A, + A,, A&. The two parallelepipeds have equal volumes.3.3 A set of axioms for a determinant function The properties of the scalar triple product mentioned in the foregoing section can besuitably generalized and used as axioms for determinants of order n. If A = (a,J is ann x n matrix with real or complex entries, we denote its rows by A,, . . . , A,, . Thus, theith row of .4 is a vector in n-space given by Ai = (ail, ai2, . . . , a,,J.We regard the determinant as a function of the n rows A,, . . . , A,, and denote its values byd(A,,..., A,) or by det A. AXIOMATIC DEFINITION OF A DETERMINANT FUNCTION. A real- or complex-valuedfunctiond, dejinedfor each ordered n-tuple of vectors A,, . . . , A, ir;l n-space, is called a determinantfunction of order n if it satisjes the following axioms for al'1 choices of vectors A,, , . . , A,and C in n-space:
74 Determinants AXIOM 1. HOMOGENEITY IN EACH ROW. If the kth row A, is multiplied by a scalar t,then the determinant is also multiplied by t: d(. . . , tA,, . . .) = td(. . . , A,, . . .). AXIOM 2. ADDITIVITY IN EACH ROW. For each k we have d(A, ,..., A,+C ,..., A,)=d(A, ,..., A, ,..., A,)+&(A, ,..., C ,..., A,). AXIOM 3. THE DETERMINANT VANISHES IF ANY TWO ROWS ARE EQUAL: d(A,,...,A,)=O if Ai = Ai for some i and j with i # j. AXIOM 4. THE DETERMINANT OF THE IDENTITY MATRIX IS EQUAL TO 1: d(I,, . . . , Z,) = 1 , where I, is the kth unit coordinate vector. The first two axioms state that the determinant of a matrix is a linear function of each ofits rows. This is often described by saying that the determinant is a multilinear function ofits rows. By repeated application of linearity in the first row we can write &,C,,A,,...,A, =~~hWwb,. . . ,A,), k=lwhere tl, . . . , t, are scalars and C,, . . . , C, are any vectors in n-space. Sometimes a weaker version of Axiom 3 is used: AXIOM 3'. THE DETERMINANT VANISHES IF TWO ADJACENT ROWS ARE EQUAL: d(A,, . . . , A,) = 0 if Ak = A,+, forsomek=1,2 ,..., n-l. It is a remarkable fact that for a given n there is one and only one function d whichsatisfies Axioms 1, 2, 3' and 4. The proof of this fact, one of the principal results of thischapter, will be given later. The next theorem gives properties of determinants deducedfrom Axioms 1,2, and 3' alone. One of these properties is Axiom 3. It should be noted thatAxiom 4 is not used in the proof of this theorem. This observation will be useful later whenwe prove uniqueness of the determinant function. THEOREM 3.1. A determinant function satisfying Axioms 1, 2, and 3' has the followingfurther properties: (a) The determinant vanishes if some row is 0: d(A,, . . .,A,)=0 if A,=0 for some k .
A set of axioms for a determinant function 75 (b) The determinant changes sign if two adjacent rows are interchanged: 4.. . , A,, &+I,. . .> = -d(. . . , AB+:,, A,, . . .). (c) The determinant changes sign if any two rows Ai and A) with i # j are interchanged. (d) The determinant vanishes if any two rows are equal: d(A,., . . .,A,)=0 i f Ai = Aj for some i and j with i # j. (e) The determinant vanishes $ its rows are dependent. Proof. To prove (a) we simply take t = 0 in Axiom 1. To prove (b), let B be a matrixhaving the same rows as A except for row k and row k + 11. Let both rows Bk and BKflbe equal to /I, + A,,, . Then det B = 0 by Axiom 3'. Thus we may write d(. . . , Ak+Ak+l,Ak+Ak+l,...)=O.Applying the additive property to row k and to row k + 1 we can rewrite this equation asfollows : d( . . . .A,,& ,... )+d(...,A,,A,+, ,... )+d( . . . .&+I,& ,...) +d( . . . .Ak+l,Ak+l,... )=O.The first and fourth terms are zero by Axiom 3'. Hence the second and third terms arenegatives of each other, which proves (b). To prove (c) we can assume that i < j . We can interchange rows Ai and Aj by performingan odd number of interchanges of adjacent rows. First we interchange row A, successivelywith the earlier adjacent rows Aj+l, Ajpz,. . . , Ai. This requires j - i interchanges.Then we interchan,ge row Ai successively with the later adjacent rows Ai+l, Ai+z, . . . , Ajpl.This requires j - i - I further interchanges. Each interchange of adjacent rows reversesthe sign of the determinant. Since there are (j - i) + (j -- i - 1) = 2(j - i) - 1 inter-changes altogether (an odd number), the determinant changes, sign an odd number of times,which proves (c). To prove (d), let B be the matrix obtained from A by interchanging rows Ai and Ad.Since Ai = Aj we have B = A and hence det B = det A. But by (c), det B = -det A.Therefore det A = 0. To prove (e) suppose scalars cr, . . . , c, exist, not all zero, such that I;=:=, ckA, = 0.Then any row A, with cK # 0 can be expressed as a linear combination of the other rows.For simplicity, suppose that A, is a linear combination of the others, say A, = zE=, t,A, .By linearity of the first row we have d(A,,A,, t.. , &Ak,A2,...,An =kt2ft 44x., A,, . . . , A,). k=2But each term d(A,, A,, . . . , A,) in the last sum is zero since A, is equal to at least one ofA 2,"', A n* Hence the whole sum is zero. If row A, is a linear combination of the otherrows we argue the same way, using linearity in the ith row. This proves (e).
Computation of determinants 77is called a diagonal matrix. Each entry ai, off the main diag,onal (i #j) is zero. We shallprove that the det.erminant of A is equal to the product of its diagonal elements,(3.5) det A = a11az2 * * - arm , The kth row of .A is simply a scalar multiple of the kth unit 'coordinate vector, A, = a,,Z, .Applying the homogeneity property repeatedly to factor out the scalars one at a time we get det A = d(Al, . . . , A,) = %A, . . . , a,,Z,J = all * * * arm d(Z,, . . . , I,).This formula can be written in the form det A = a,, - - * a,,,, det I,where Z is the identity matrix. Axiom 4 tells us that det Z = 1 so we obtain (3.5). EXAMPLE 3. Determinant of an upper triangular matrix. A square matrix of the formis called an upper triangular matrix. All the entries below the main diagonal are zero. Weshall prove that the determinant of such a matrix is equal to the product of its diagonalelements, det U = ullu22 * * * u,, . First we prove that det U = 0 if some diagonal element Q = 0. If the last diagonalelement unn is zero., then the last row is 0 and det U = 0 by Theorem 3.1 (a). Suppose, then,that some earlier diagonal element uii is zero. To be specific, say uz2 = 0. Then each ofthe n - 1 row-vectors U,, . . . , U, has its first two components zero. Hence these vectorsspan a subspace of dimension at most n - 2. Therefore these n - 1 rows (and hence allthe rows) are dependent. By Theorem 3.1(e), det U = 0. In the same way we find thatdet U = 0 if any diagonal element is zero. Now we treat the general case. First we write the first row UL as a sum of two row-vectors,where V, = [ull, 0, . . . , 0] and Vi = [O, Q, u,,, . . . , uln]. By linearity in the first rowwe have det. U = det (V,, U,, . . . , U,) + det (Vi, U2,. . . , U,).
78 DeterminantsBut det (Vi, U,, . . . , U,) = 0 since this is the determinant of an upper triangular matrixwith a diagonal element equal to 0. Hence we have(3.6) detU=det(T/,,U, ,..., U,,).Now we treat row-vector U, in a similar way, expressing it as a sum,where v2 = LO, u22,0, * * . 3 01 and v; = [o, O, u23, . . . , UZn] *We use this on the right of (3.6) and apply linearity in the second row to obtain(3.7) det U = det (V,, V,, U,, . , . , U,,),since det (V,, Vi, U3, . . . , U,) = 0. Repeating the argument on each of the succeedingrows in the right member of (3.7) we finally obtain detU=det(VI,V, ,..., V,),where (V,, V,, . . . , V,) is a diagonal matrix with the same diagonal elements as U. There-fore, by Example 2 we have det U = ullu22 . + * u,,,as required. EXAMPLE 4. Computation by the Gauss-Jordan process. The Gauss-Jordan eliminationprocess for solving systems of linear equations is also one of the best methods for com-puting determinants. We recall that the method consists in applying three types of operationsto the rows of a matrix: (1) Interchanging two rows. (2) Multiplying all the elements in a row by a nonzero scalar. (3) Adding to one row a scalar multiple of another.By performing these operations over and over again in a systematic fashion we can transformany square matrix A to an upper triangular matrix U whose determinant we now knowhow to compute. It is easy to determine the relation between det A and det U. Each timeoperation (1) is performed the determinant changes sign. Each time (2) is performed witha scalar c # 0, the determinant is multiplied by c. Each time (3) is performed the deter-minant is unaltered. Therefore, if operation (1) is performedp times and if c1 , . . . , C~ arethe nonzero scalar multipliers used in connection with operation (2), then we have(3.8) det A = (- 1)P(c,c2 * * * c,)-l det U.Again we note that this formula is a consequence of the first three axioms alone. Its proofdoes not depend on Axiom 4.
The product formula for determinants 817. State and prove a generalization of Exercise 6 for the determinant prove that F'(x) = I;,!:) ii::)1. (b) State and prove a corresponding result for 3 x 3 determinants, assuming the validity of Equation (3.2). 9. Let U and V be two 12 x n upper triangular matrices. (a) Prove that each of U + Vand UV is an upper triangular matrix. (h) Prove that det (UV) = (det U)(det V) . (c) If det U # 0 prove that there is an upper triangular matrix CJ-l such that UU-l = I, and deduce that det (U-l) = l/det U.10. Calculate det '4, det (A-l), and 4-l for the following upper triangular matrix: I 1 2 3 4 5 0 2 3 4 0 0 2 3 ' A = 0 0 0 23.7 The product formula for determinants In this section 'we use the uniqueness theorem to prove that the determinant of a productof two square matrices is equal to the product of their determinants, det (AB) = (det A)(det B) ,assuming that a determinant function exists. We recall that the product AB of two matrices A = (a,,) and B = (bij) is the matrixC = (cij) whose i, j entry is given by the formula 12(3.12) cii = C aikbkj. k=lThe product is de:fined only if the number of columns of the left-hand factor A is equal tothe number of rows of the right-hand factor B. This is always the case if both A and B aresquare matrices of the same size. The proof of the product formula will make use of a simple relation which holds betweenthe rows of AB a:nd the rows of A. We state this as a lemma. As usual, we let Ai denotethe ith row of matrix A.
Determinants and independence of vectors 83But now it is a simple matter to verify that f satisfies Axioms 1, 2, and 3 for a determinantfunction so, by the uniqueness theorem, Equation (3.13) holds for every matrix A. Thiscompletes the proof. Applications of the product formula are given in the next two sections.3.8 The determinant of the inverse of a nonsingular matrix We recall that a. square matrix A is called nonsingular if it has a left inverse B such thatBA = I. If a left inverse exists it is unique and is also a right inverse, AB = I. We denotethe inverse by A-l. The relation between det A and det A-' is as natural as one could expect. THEOREM 3.5. If matrix A is nonsingular, then det A # 0 and we have(3.14) &t A-l = 1 det A ' Proqfi From the product formula we have (det A)(det A-l) = det (AA-') = det .I = 1.Hence det A # 0 and (3.14) holds. Theorem 3.5 shows that nonvanishing of det A is a necessary condition for A to be non-singular. Later we will prove that this condition is also sufficient. That is, if det A # 0then A--l exists.3.9 Determinants and independence of vectors A simple criterion for testing independence of vectors can be deduced from Theorem 3.5. THEOREM 3.6. A set of n vectors A,, . . . , A, in n-space is independent if and only ifW, , . . . , A,) # 0. Proof. We already proved in Theorem 3.2(e) that dependence implies d(A, , . . . , A,) =0. To prove the converse, we assume that A,, . . . , A, are independent and prove thatd(A,, . . . , A,) # 0. Let Vn denote the linear space of n-tuples of scalars. Since A', , . . . , A, are n independentelements in an n-dimensional space they form a basis for V, . By Theorem 2.12 there is alinear transformation T: V, + V, which maps these n vectors onto the unit coordinatevectors, TV,) = Ire for k=l,2 ,..., n.Therefore there is an n x n matrix B such that A,B = I, for k=l,2 ,..., n.
84 DeterminantsBut by Lemma 3.3 we have A,B = (AB), , where A is the matrix with rows A,, . . . , A,.Hence AB = I, so A is nonsingular and det A # 0.3.10 The determinant of a block-diagonal matrix [1 A square matrix C of the form A 0 c = 0 B'where A and B are square matrices and each 0 denotes a matrix of zeros, is called a block-diagonal matrix with two diagonal blocks A and B. An example is the 5 x 5 matrix 0 0 4 5 6 10The diagonal blocks in this case are A= [1 0 1 and The next theorem shows that the determinant of a block-diagonal matrix is equal to theproduct of the determinants of its diagonal blocks. THEOREM 3.7. For any two square matrices A and B we have(3.15) det = (det A)(det B). Proof. AssumeAisnxnandBismxm. We note that the given block-diagonalmatrix can be expressed as a product of the formwhere I,, and I,,, are identity matrices of orders n and m, respectively. Therefore, by theproduct formula for determinants we have(3.16) det[t I]=det[t ZI]det[z Og].
--- Exercises 85 [ I A 0Now we regard the determinant det as a function of the n rows of A. This is 0 Anpossible because of the block of zeros in the upper right-hand corner. It is easily verifiedthat this function satisfies all four axioms for a determinant function of order n. Therefore,by the uniqueness theorem, we must have det = det A.A similar argument shows that det = det B. Hence (3.16) implies (3.15).3.11 Exercises1. For each of the rfollowing statements about square matrices, give a proof or exhibit a counter example. (a) det(A +B) =detA +detB. (b) det {(A + B)12} = {det (A + B)}2 (c) det {(A + B)2} = det (A2 + 2AB + B2) (d) det {(A + B)2} = det (A2 + B2).2. (a) Extend 'Theorem 3.7 to block-diagonal matrices with three diagonal blocks: = (det A)(det B)(det C). (b) State and prove a generalization for block-diagonal matrices with any number of diagonal blocks.3.L,etA=[ i! i, B=i i 21. ProvethatdetA=det[i t]andthat [1 a $b det B = det e .f '4. State and prove a generalization of Exercise 3 for n x n matrices. a b 0 0 c d 0 05,LetA= ProvethatdetA=det[I z]detK L]. efgh' I x y .z w16. State and prove a generalization of Exercise 5 for II x n matrices of the form A = where B, C, D denote square matrices and 0 denotes a matrix of zeros.
86 Determinants7. Use Theorem 3.6 to determine whether the following sets of vectors are linearly dependent or independent. (a) A, = (1, -l,O), A, = (O,l, -l), A, = (2,3, -1). (b)A,=(1,-1,2,1),A,=(-1,2,-1,O),Ag=(3,-1,1,0),A~=(1,0,0,1). Cc) A,=(1,0,0,0,1),A,=(1,1,0,0,O),As=(1,0,1,0,1),Aq=(1,1,0,1,1), A, = (O,l,O, 1,O).3.12 Expansion formulas for determinants. Minors and cofactors We still have not shown that a determinant function actually exists, except in the 2 x 2case. In this section we exploit the linearity property and the uniqueness theorem to showthat if determinants exist they can be computed by a formula which expresses every deter-minant of order n as a linear combination of determinants of order n - 1 . Equation (3.2)in Section 3.1 is an example of this formula in the 3 x 3 case. The general formula willsuggest a method for proving existence of determinant functions by induction. Every row of an n x n matrix A can be expressed as a linear combination of the n unitcoordinate vectors II, . . . , Z, . For example, the first row A, can be written as follows: A, = $a,J, . 61Since determinants are linear in the first row we have ia,$,,A, ,..., A, =$;u d(l,, AZ,. . . , A,). j=lTherefore to compute det A it suffices to compute d(Z,, A,, . . . , A,) for each unit coordinatevector Zj . Let us use the notation ,4ij to denote the matrix obtained from A by replacing the firstrow A, by the unit vector Zj . For example, if IZ = 3 there are three such ma&ices:Note that det Aij = d(Zj, AZ, . . . , A,). Equation (3.17) can now be written in the form(3.18) det A = i all det Ali. j=lThis is called an expansion formula; it expresses the determinant of A as a linear combina-tion of the elements in its first row. The argument used to derive (3.18) can be applied to the kth row instead of the first row.The result is an expansion formula in terms of elements of the kth row.
Expansion formulas for determinants. Minors and cofactors 87 THEOREM 3.8. EXPANSION BY COFACTORS. Let Akj denote the matrix obtained from A by replacing the kth row A, by the unit coordinate vector Ij. Then we have the expansionformula(3.19) det A = takj det Akj j,lwhich expresses the determinant of A as a linear combination of the elements of the kth row.The number det Akj is called the cofactor of entry akj. In the next theorem we shall prove that each cofactor is, except for a plus or minus sign,equal to a determinant of a matrix of order n - 1 . These smaller matrices are calledminors. DEFINITION. Given a square matrix A of order n 2 2, the square matrix of order n - 1obtained by deleting the kth row and the jth column of A is called the k, j minor of A and isdenoted by A,. . EXAMPLE . A matrix A = (alei) of order 3 has nine minors. Three of them are &I=[;;: ;;;]y Alz=[;:: ;;l]r &a=[;;; ;:]- Equation (3.2) expresses the determinant of a 3 x 3 matrix as a linear combination ofdeterminants of these three minors. The formula can be written as follows: det A = a,, det A,, - aI det A,, + aI3 det A,, .The next theorem extends this formula to the n x n case for any n 2 2. THEOREM 3.9. EXPANSION BY kTH-ROW MINORS. For any n x n matrix A, n > 2, thecofactor of ski is related to the minor Aki by the formula(3.20) det ALj = (- l)kfj det Aki.Therefore the expansion of det A in terms of elements of the kth row is given by(3.21) det A = $( - l)"+jakj det A,$. j=l Proof. We illustrate the idea of the proof by considering first the special case k = j = 1.The matrix ,4& has the form 1 0 0 *a* o- azl az2 az3 *** azn a31 a32 a33 * *a a3n A;, = .a,, an2 an3 * * * an+
88 Determinants-By applying elementary row operations of type (3) we can make every entry below the 1 inthe first column equal to zero, leaving all the remaining entries intact. For example, if wemultiply the first row of Ai, by -azl and add the result to the second row, the new secondrow becomes (0, az2, az3, . . . , a2J. By a succession of such elementary row operationswe obtain a new matrix which we shall denote by A& and which has the form 0 a22 a32Since row operations of type (3) leave the determinant unchanged we have(3.22) det A!, = det A;, .But A,O, is a block-diagonal matrix so, by Theorem 3.7, we have det A!, = det AlI,where A,, is the 1, 1 minor of A,Therefore det Ai, = det AlI, which proves (3.20) for k = j = 1. We consider next the special case k = 1 ,i arbitrary, and prove that(3.23) det Aij = (- l)j-' det A,$.Once we prove (3.23) the more general formula (3.20) follows at once because matrix AJ,can be transformed to a matrix of the form Bij by k - 1 successive interchanges of adjacentrows. The determinant changes sign at each interchange so we have(3.24) det Akj = (- l)"-1 det B&,where B is an n x n matrix whose first row is Ij and whose 1 ,i minor BIj is Akj. By (3.23),we have det Bij = (- l)'-l det Blj = (- l)j-' det Aki,so (3.24) gives us det ALj = (- l)"-l( - l)j-' det Akj = (- l)"+j det Akj.Therefore if we prove (3.23) we also prove (3.20).
90 Determinants The entries along the sloping lines are all 1. The remaining entries not shown are all 0.By interchanging the first row of C successively with rows 2, 3, . . . ,j we arrive at the n x nidentity matrix I after j - 1 interchanges. The determinant changes sign at each inter-change, so det C = (- l)j-'. Hencef(J) = (- l)j-', which proves (3.23) and hence (3.20).3.13 Existence of the determinant function In this section we use induction on n, the size of a matrix, to prove that determinantfunctions of every order exist. For n = 2 we have already shown that a determinant functionexists. We can also dispense with the case n = 1 by defining det [aa,,] = a,, . Assuming a determinant function exists of order n - 1, a logical candidate for a deter-minant function of order n would be one of the expansion formulas of Theorem 3.9, forexample, the expansion in terms of the first-row minors. However, it is easier to verify theaxioms if we use a different but analogous formula expressed in terms of the first-columnminors. THEOREM 3.10. Assume determinants of order n - 1 exist. For any n X n matrixA = (aik) , let f be the function dejned by the formula(3.26) f(A,, . . . , A,) = j=l l)j+'aj, det Aj,. i(-Then f satisjes allfour axiomsfor a determinantfunction of order n. Therefore, by induction,determinants of order n exist for every n. Proof. We regard each term of the sum in (3.26) as a function of the rows of A and wewrite .fW,,.. . , A,) = (- l)jflaj,det Aj, .If we verify that eachf, satisfies Axioms 1 and 2 the same will be true forf. Consider the effect of multiplying the first row of A by a scalar t. The minor A,, is notaffected since it does not involve the first row. The coefficient a,, is multiplied by t, so wehave f,(tA, 9 A,z 3 . . . > A,) = tall det A,, = tf,(A,, . . . , A,).Ifj > 1 the first row of each minor Aj, gets multiplied by t and the coefficient ajI is notaffected, so again we have &@A,, A,, . . . , A,) = tY&h, A,, . . . , A,).Therefore each fj is homogeneous in the first row. If the kth row of A is multiplied by t, where k > 1, the minor A,, is not affected but akl ismultiplied by t, so fk is homogeneous in the kth row. If j # k, the coefficient aj, is notaffected but some row of Aj, gets multiplied by t. Hence every J; is homogeneous in thekth row.
The determinant of a transpose 91 A similar argument shows that each fi is additive in every row, so f satisfies Axioms 1 and2. We prove next that f satisfies Axiom 3', the weak version of Axiom 3. From Theorem 3.1, it then follows that f satisfies Axiom 3. To verify that. f satisfies Axiom 3', assume two adjacent rows of A are equal, say A, =A k+l Then, except for minors Akl and A,+, 1, each mi,nor A), has two equal rows so det .4,, = 0. Therefore the sum in (3.26) consists only of the two terms corresponding toJ'=kand,i=k+ 1,(3.27) f(A,>..., An) = (-l)k+lakl det A,, + (-1 )"i-2ak+,,, det A,,,,, .But A,, = A,,,,, and akl = a,,,,, since A, = A,,, . Therefore the two terms in (3.27)differ only in sign, so f (A,, . . . , A,) = 0. Thus, f satisfies Axiom 3'. Finally, we verify that f satisfies Axiom 4. When A = 1 we have a,, = 1 and a,, = 0 fori > 1. Also, A,, is the identity matrix of order n - 1, so each term in (3.26) is zero exceptthe first, which is equal to 1. Hence f (f, , . . . , 1,) = 1 sgof satisfies Axiom 4. In the foregoing proof we could just as well have used a. functionfdefined in terms of thekth-column minors Aik instead of the first-column minors Aj, . In fact, if we let(3.28) f(Al,..., A,) = 2 (- l)ii-kajl, det Ajk , j=lexactly the same type of proof shows that this f satisfies all four axioms for a determinantfunction. Since determinant functions are unique, the expansion formulas in (3.28) andthose in (3.21) are all equal to det A. The expansion formulas (3.28) not only establish the existence of determinant functionsbut also reveal a new aspect of the theory of determinants-a connection between row-properties and column-properties. This connection is discussed further in the next section.3.14 The determinant of a transpose Assoc:iated with each matrix A is another matrix called the transpose of A and denotedby At. The rows of At are the columns of A. For example, if A =A formal definition may be given as follows. DEFI'VITION OF TRANSPOSE. The transpose of an m X: n matrix A = (a&~& is the n X mmatrix At whose i, j entry is aji . Although transposition can be applied to any rectangular matrix we shall be concernedprimarily with square matrices. We prove next that transposition of a square matrix doesnot alter its determinant,
92 Determinants THEOREM 3. Il. For any n x n matrix A we have det A = det At. Proof. The proof is by induction on n. For n = 1 and n = 2 the result is easily verified. Assume, then, that the theorem is true for matrices of order n - 1 . Let A = (aij) and let B = At = (6,). Expanding det A by its first-column minors and det B by its first-row minors we have det A = &-l)'flaj, det Ajl, det B = i (- l)'+'blj det Blj. kl i=lBut from the definition of transpose we have bli = a,, and Blj = (AJ". Since we areassuming the theorem is true for matrices of order n - 1 we have det Blj = det Aj,.Hence the foregoing sums are equal term by term, so det A = det B.3.15 The cofactor matrix Theorem 3.5 showed that if A is nonsingular then det A # 0. The next theorem provesthe converse. That is, if det A # 0, then A-l exists. Moreover, it gives an explicit formulafor expressing A-l in terms of a matrix formed from the cofactors of the entries of A. In Theorem 3.9 we proved that the i, j cofactor of aij is equal to (- l)i+i det Aii , whereAii is the i, j minor of A. Let us denote this cofactor by cof aij. Thus, by definition, cof aij = (- l)ifjdet Aij . DEFINITION OF THE COFACTOR MATRIX. The matrix whose i, j entry is cof aii is called thecofactor matrix? of A and is denoted by cof A. Thus, we have cof A = (cof aij)& = (( - l)i+i det Ai,)tj=, . The next theorem shows that the product of A with the transpose of its cofactor matrix is,apart from a scalar factor, the identity matrix I. THEOREM 3.12. For any n x n matrix A with n 2 2 we have(3.29) A(cof A)t = (det A)Z.In particular, if det A # 0 the inverse of A exists and is given by A-l = bA (cof A)t.t In much of the matrix literature the transpose of the cofactor matrix is called the adjugate of A. Some ofthe older literature calls it the adjoint of A. However, current nomenclature reserves the term adjoint foran entirely different object, discussed in Section 5.8.
Cramer.? rule 93 Proof. Using Theorem 3.9 we express det A in terms of its kth-row cofactors by theformula(3.30) det A = i akj cof akj. j=lKeep k fixed and apply this relation to a new matrix B whose ith row is equal to the kthrow of A for some i # k, and whose remaining rows are the same as those of A. Thendet B = 0 because the ith and kth rows of B are equal. Expressing det B in terms of itsith-row cofactors we have(3.31) det B = 5 bij cof bij = 0. j=lBut since the ith row of B is equal to the kth row of A wc have bii = akj and cof bij = cof aij for every j .Hence (3.31) states that(3.32) if k + i.Equations ((3.30) and (3.32) together can be written as follows: 12(3.33) c j=l akj cof aij = det A 0 if i = k i f i#k.But the sum appearing on the left of (3.33) is the k, i entry of the product A(cof #. There-fore (3.33) implies (3.29). As a direct corollary of Theorems 3.5 and 3.12 we have the following necessary andsufficient condition for a square matrix to be nonsingular. THEOREM 3.13. A square matrix A is nonsingular if and <only if det A # 0.3.16 Cramer's rule Theorem 3.12 can also be used to give explicit formulas for the solutions of a system oflinear equations with a nonsingular coefficient matrix. The formulas are called Cramer'srule, in honor of the Swiss mathematician Gabriel Cramer (1704-1752). THEOREM 3.14. CRAMERS RULE. If a system of n &near equations in n unknowns
94 Determinantshas a nonsingular coeficient-matrix A = (aij) , then there is a unique solution for the system&en by the formulas 1 n(3.34) b, cof akj , for j=l,2 ,..., n. xj =det c k=l Proof. The system can be written as a matrix equation, ' AX=B,where X and B are column matrices, X = Since A is nonsingularthere is a unique solution X given by(3.35) X = A-'B = bA (cof A)tB.The formulas in (3.34) follow by equating components in (3.35). It should be noted that the formula for xj in (3.34) can be expressed as the quotient of twodeterminants, x.=2, det C. I det Awhere Cj is the matrix obtained from A by replacing the jth column of A by the columnmatrix B.3.17 Exercises1. Determine the cofactor matrix of each of the following matrices :2. Determine the inverse of each of the nonsingular matrices in Exercise 1.3. Find all values of the scalar A for which the matrix II - A is singular, if A is equal to
Exercises 954. If A is an n x n matrix with n 2 2, prove each of the following, properties of its cofactor matrix: (a) cof (At) = (cof #. (b) (cof #A = (det A)Z. (c) A(cof ~1)~ = (cof #A (A commutes with the transpose of its cofactor matrix).5. Use Cramer's rule to solve each of the following systems: (a) x + 5 + 3.z = 8, 2 x - y +4z =7, -y + z = 1. (b) x +y +2z =O, 3x-y-z=3, 2x +5y +3z =4.6. (a) Explain why each of the following is a Cartesian equation for a straight line in the xy-plane passing thtough two distinct points (x1, yl) and (x2, yz). (b) State and prove corresponding relations for a plane in 3-space passing through three distinct points. (c) State alnd prove corresponding relations for a circle in the xy-plane passing through three noncolinear points.7. Given n2 functions fij, each differentiable on an interval (a, i5), define F(x) = det [j&x)] for each x in (a, b). Prove that the derivative F'(x) is a sum of n determinants, F'(x) = 2 det A,(x), i=l where Ai is the matrix obtained by differentiating the fun'ctions in the ith row of [f&(x)].8. An n x n matrix of functions of the form W(x) = [uji-l'(x)], in which each row after the first is the derivative Iof the previous row, is called a Wronskian matrix in honor of the Polish mathe- matician J. M. II. Wronski (1778-1853). Prove that the derivative of the determinant of W(x) is the determinant of the matrix obtained by differentiating each entry in the last row of W(X). [Hint: Use Exercise 7.1
4 EIGENVALUES AND EIGENVECTORS4.1 Linear transformations with diagonal matrix representations Let T: V+ V be a linear transformation on a finite-dimensional linear space V. Thoseproperties of T which are independent of any coordinate system (basis) for V are calledintrinsicproperties of T. They are shared by all the matrix representations of T. If a basiscan be chosen so that the resulting matrix has a particularly simple form it may be possibleto detect some of the intrinsic properties directly from the matrix representation. Among the simplest types of matrices are the diagonal matrices. Therefore we might ask whether every linear transformation has a diagonal matrix representation. In Chapter 2 we treated the problem of finding a diagonal matrix representation of a linear transfor-mation T: V+ W, where dim V = n and dim W = m . In Theorem 2.14 we proved that there always exists a basis (el, . . . , e,) for V and a basis (M.', , . . . , w,) for W such that the matrix of T relative to this pair of bases is a diagonal matrix. In particular, if W = V the matrix will be a square diagonal matrix. The new feature now is that we want to use thesame basis for both Vand W. With this restriction it is not always possible to find a diagonalmatrix representation for T. We turn, then, to the problem of determining which trans- formations do have a diagonal matrix representation. Notation: If A = (aij) is a diagonal matrix, we write A = diag (all, az2, . . . , a,,). It is easy to give a necessary and sufficient condition for a linear transformation to have adiagonal matrix representation. THEOREM 4.1. Given a linear transformation T: V--f V, where dim V = n. If T has adiagonal matrix representation, then there exists an independent set o f elements ul, . . . , u,in V and a corresponding set o f scalars A,, . . . , A, such that(4.1) T&J = &curt for k=1,2 ,..., n.Conversely, i f there is an independent set ul, . . . , u, in V and a corresponding set o f scalarsA I,"', 1, satisfying (4.1), then the matrix A =diag(&,...,&)is a representation o f T relative to the basis (uI , . . . , u,).96
Eigenvectors and eigenvalues of a linear transformation 97 Proof. Assume first that T has a diagonal matrix representation A = (ai& relative tosome basis (e,, . . . , e,). The action of T on the basis elements is given by the formula T(eJ = i aikei = akBek i=lsince aik = 01 for i Z k . This proves (4.1) with uk = ek and 1, = a,, . Now suppose independent elements ul, . . . , U, and scalars &, . . . , A, exist satisfying(4.1). Since zfl, . . . , u, are independent they form a basis for V. If we define a,, = 1, andaik = 0 for i # Ic, then the matrix A = (a& is a diagonal matrix which represents Trelative to the basis (ul, . . . , u,). Thus the probkm of finding a diagonal matrix representation of a linear transformationhas been transformed to another problem, that of finding independent elements ul, . . . , U,and scalars ;I,, . . . , 1, to satisfy (4.1). Elements uk and scalars 1, satisfying (4.1) are calledeigenvectors and eigenvalues of T, respectively. In the next section we study eigenvectorsand eigenvaluesf in a more general setting.4.2 Eigenvectors and eigenvalues of a linear transformation In this discussion V denotes a linear space and S denotes a subspace of V. The spaces Sand V are not required to be finite dimensional. DEFINITION. Let T: S 4 V be a linear transformation of S into V. A scalar ;i is called aneigenvalue of' T if'there is a nonzero element x in S such that(4.2) T(x) = Ix.The element .x is called an eigenvector of Tbelonging to 1. The scalar 1 is called an eigenvaluecorresponding to .x. There is exactly one eigenvalue corresponding to a given eigenvector x. In fact, if wehave T(x) = iix and T(x) = ,UX for some x # 0, then 1x = yx so il = ,u . q Note: Although Equation (4.2) always holds for x = 0 and any scalar I, the definition excludes 0 as an eigenvector. One reason for this prejudice against 0 is to have exactly one eigenvalue ), associated with a given eigenvector x. The following examples illustrate the meaning of these concepts. EXAMPLE 1. Multiplication by a fixed scalar. Let T: S + V be the linear transformationdefined by the equation T(x) = cx for each x in S, where c is a fixed scalar. In this exampleevery nonzero element of S is an eigenvector belonging to the scalar c.t The words eigenvector and eigenvalue are partial translations of the German words Egenvektor andEigenwert, respectively. Some authors use the terms characteristic vector, or proper vector as synonyms foreigenvector. Eigenv,alues are also called characteristic values, proper values, or latent roots.
98 Eigenvalues and eigenvectors EXAMPLE 2. The eigenspace E(A) consisting of all x such that T(x) = Ax. Let T: S -+ Vbe a linear transformation having an eigenvalue 1. Let E(A) be the set of all elements x inS such that T(x) = 1.~. This set contains the zero element 0 and all eigenvectors belongingto il. It is easy to prove that E(I) is a subspace of S, because if x and y are in E(1) we have T(aX + by) = aT(x) + bT(y) = ailx + bAy = A(ax + by)for all scalars a and b. Hence (ax + by) E E(1) so E(A) is a subspace. The space E(1) iscalled the eigenspace corresponding to 1. It may be finite- or infinite-dimensional. If E(A)is finite-dimensional then dim E(I) 2 1 , since E(1) contains at least one nonzero element xcorresponding to 1. EXAMPLE 3. Existence of zero eigenvalues. If an eigenvector exists it cannot be zero, bydefinition. However, the zero scalar can be an eigenvalue. In fact, if 0 is an eigenvalue forx then T(x) = Ox = 0, so x is in the null space of T. Conversely, if the null space of Tcontains any nonzero elements then each of these is an eigenvector with eigenvalue 0. Ingeneral, E(A) is the null space of T - AZ. EXAMPLE 4. Rejection in the xy-plane. Let S = V = V,(R) and let T be a reflection inthe xy-plane. That is, let Tact on the basis vectors i, j, k as follows: T(i) = i, T(j) = j,T(k) = -k. Every nonzero vector in the xy-plane is an eigenvector with eigenvalue 1.The remaining eigenvectors are those of the form ck, where c # 0 ; each of them haseigenvalue - 1 . EXAMPLE 5. Rotation of theplane through ajxedangle u. This example is of special interestbecause it shows that the existence of eigenvectors may depend on the underlying field ofscalars. The plane can be regarded as a linear space in two different ways: (1) As a 2-dimensional real linear space, V = V,(R), with two basis elements (1, 0) and (0, l), andwith real numbers as scalars; or (2) as a l-dimensional complex linear space, V = V,(C),with one basis element 1, and complex numbers as scalars. Consider the second interpretation first. Each element z # 0 of V,(C) can be expressedin polar form, z = yei*. If T rotates z through an angle cc then T(z) = reicefa) = eiaz.Thus, each z # 0 is an eigenvector with eigenvalue il = eia. Note that the eigenvalue ei" isnot real unless c4 is an integer multiple of r. Now consider the plane as a real linear space, V,(R). Since the scalars of V,(R) are realnumbers the rotation T has real eigenvalues only if tc is an integer multiple of n. In otherwords, if CI is not an integer multiple of v then T has no real eigenvalues and hence noeigenvectors. Thus the existence of eigenvectors and eigenvalues may depend on the choiceof scalars for V. EXAMPLE 6. The dzyerentiation gperator. Let V be the linear space of all real functions fhaving derivatives of every order on a given open interval. Let D be the linear transfor-mation which maps each f onto its derivative, D(f) = f '. The eigenvectors of D are thosenonzero functions f satisfying an equation of the form
Eigenvectors and eigenvalues of a linear transformation 99 f' = Iffor some real 1. This is a first order linear differential equation. All its solutions are givenby the formula f(x) = ce"",where c is an arbitrary real constant. Therefore the eigenvectors of D are all exponentialfunctions f(x) == ceAx with c # 0 . The eigenvalue corresponding to f(x) = celZ is 1. Inexamples li'ke this one where V is a function space the eigenvectors are called eigenfunctions. EXAMPLE 7. The integration operator. Let V be the linear space of all real functionscontinuous on a finite interval [a, b]. Iffg V define g = T(f) to be that function given by g(x) = sazf(t) dt i f a<x<b.The eigenfunctions of T (if any exist) are those nonzerofsatisfying an equation of the form(4.3) I; f(t) dt = Y(x)for some real 4. If an eigenfunction exists we may differentiate this equation to obtain therelationf(x) = 3tf'(x), from which we findf(x) = ce"'", provided 3, # 0. In other words,the only candidates for eigenfunctions are those exponential functions of the formf(x) =cexlk with c # 0 and 1 # 0. However, if we put x = a in (4.3) we obtain 0 = If(a) = Ice"'".Since call is never zero we see that the equation T(f) = @-cannot be satisfied with a non-zero f, so Thas no eigenfunctions and no eigenvalues. EXAMPLE 8. The subspace spanned by an eigenvector. Let T: S + V be a linear trans-formation having an eigenvalue 1. Let x be an eigenvector belonging to 3, and let L(x) bethe subspace spanned by x. That is, L(x) is the set of all scalar multiples of x. It is easy toshow that T maps L(x) into itself. In fact, if y = cx we Ihave T(y) = T(cx) = CT(X) = c(Ax) = l(cx) = Ay.If c # 0 then y # 0 so every nonzero element y of L(x) is also an eigenvector belongingto 1. A subspace U of S is called invariant under Tif Tmaps each element of U onto an elementof U. We have j ust shown that the subspace spanned by an eigenvector is invariant under T.
100 Eigenvalues and eigenvectors4.3 Linear independence of eigenvectors corresponding to distinct eigenvalues One of the most important properties of eigenvalues is described in the next theorem. Asbefore, S denotes a subspace of a linear space V. THEOREM 4.2. Let ul, . . . , uk be eigenvectors of a linear transformation T: S + V, andassume that the corresponding eigenvalues A,, . . . , ii, are distinct. Then the eigenvectors%, * *. 9 uk are independent. Proof. The proof is by induction on k. The result is trivial when k = 1 . Assume, then,that it has been proved for every set of k - 1 eigenvectors. Let ul, . . . , uk be k eigen-vectors belonging to distinct eigenvalues, and assume that scalars ci exist such that(4.4) iciui = O *Applying T to both members of (4.4) and using the fact that T(uJ = ii,u, we findMultiplying (4.4) by 1., and subtracting from (4.5) we obtain the equation k-l isCi(A - 'kj"i = 0 *But since ul, . . . , uk-l are independent we must have ci(ili - jlk) = 0 for each i = 1, 2, . . . ,k - 1 . Since the eigenvalues are distinct we have ai # )Lk for i # k so ci = 0 for i = 1, 2,. . . ) k - 1 . From (4.4) we see that ck is also 0 so the eigenvectors ul, . . . , uk are inde-pendent. Note that Theorem 4.2 would not be true if the zero element were allowed to be an eigen-vector. This is another reason for excluding 0 as an eigenvector. Warning: The converse of Theorem 4.2 does not hold. That is, if T has independent eigenvectors ul, . . . , uk, then the corresponding eigenvalues A,, . . . , 2, need not be dis- tinct. For example, if T is the identity transformation, T(x) = x for all x, then every x # 0 is an eigenvector but there is only one eigenvalue, ii = 1 . Theorem 4.2. has important consequences for the finite-dimensional case. THEOREM 4.3. If dim V = n , every linear transformation T: V + V has at most n distincteigenvalues. If T has exactly n distinct eigenvalues, then the corresponding eigenvectors forma basis for V and the matrix of T relative to this basis is a diagonal matrix with the eigenvaluesas diagonal entries. Proof. If there were n + 1 distinct eigenvalues then, by Theorem 4.2, V would containn + 1 independent elements. This is not possible since dim V = n . The second assertionfollows from Theorems 4.1 and 4.2.
Exercises 101 Note: Theorem 4.3 tells us that the existence of n distinct eigenvalues is a s@cient condition for Tto have a diagonal matrix representation. This condition is not necessary. There exist linear transformations with less than n distinct eigenvalues that can be represented by diagonal matrices. The identity transformation is an example. All its eigen- values are eqiual to 1 but it can be represented by the identity matrix. Theorem 4.1 tells us that the existence of n independent eigenvectors is necessary and sufJient for T to have a diagonal matrix representation.4.4 Exercises 1. (a) If T has an eigenvalue 1, prove that aT has the eigenvalue al. (b) If x is an eigenvector for both Tl and T, , prove that x is also an eigenvector for aT, + bT, . How are the eigenvalues related? 2. Assume 'T: V-- Vhas an eigenvector x belonging to an eigenvalue 1. Prove that xGs an eigen- vector of T2 belonging to A2 and, more generally, x is an eigenvector of Tn belonging to 2". Then use the result of Exercise 1 to show that if P is a polynomial, then x is an eigenvector of P(T) belongin,g to P(A). 3. Consider the plane as a real linear space, V = V,(R) , and let T be a rotation of V through an angle of ~12 radians. Although T has no eigenvectors, prove that every nonzero vector is an eigenvector for T2. 4. If T: V--c V has the property that T2 has a nonnegative eigenvalue A2, prove that at least one of 3, or --3, is an eigenvalue for T. [Hint: T2 - A21 = (T + lZ)( T - AZ) .] 5. Let V be the linear space of all real functions differentiable on (0, 1). If f E V, define g = T(f) to mean i:hatg(t) = tf '(t) for all t in (0, 1). Prove that every real 1 is an eigenvalue for T, and determine the eigenfunctions corresponding to A. 6. Let V be the linear space of all real polynomials p(x) of degree < n. If p E V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue? 7. Let V be the linear space of all functions continuous on ( - 00, + to) and such that the integral J?., f(t) dt exists for all real x. If f E V, let g = T(f) be defined by the equation g(x) = 5". co f (t) dt . F'rove that every positive I is an eigenvalue for Tand determine the eigenfunctions corresponding to 1. 8. Let V be lthe linear space of all functions continuous on ( - ~1, + a) and such that the integral I"., t f(t) dt exists for all real x. If f 6 Y let g = T(f) be defined by the equationg(x) = JY m t f (t) dt . Prove that every negative 1 is aneigenvalue for T and determine the eigenfunc- tions corresponding to 1. 9. Let V = C(0, n) be the real linear space of all real functions continuous on the interval [0, ~1. Let S be the subspace of all functions f which have a continuous second derivative in linear and which also satisfy the boundary conditions f(0) =f"(r) = 0. Let T: S + V be the linear transformation which maps each f onto its second derivative, T(f) = f II. Prove that the eigenvalues of Tare the numbers of the form -n2, where n = 1, 2, . . . , and that the eigen- functions corresponding to -n2 are f(t) = c, sin nt , where L', # 0.10. Let V be the linear space of all real convergent sequences {x,}. Define T: V---t V as follows: If x = {x,} is a convergent sequence with limit a, let T(x) = {y,}, where yS = a - x, for n 2 1 . Prove that Thas only two eigenvalues, 1 = 0 and 3, = -1, and determine the eigen- vectors belonging to each such 1.11. Assume that a linear transformation T has two eigenvectors x and y belonging to distinct eigenvalues i, and p. If ax + by is an eigenvector of T, prove that a = 0 or b = 0.12. Let T: S -+ Vbe a linear transformation such that every nonzero element of S is an eigenvector. Prove that therle exists a scalar c such that T(x) = cx . In other words, the only transformation with this property is a scalar times the identity. [Hint: Use Exercise 11.1
102 EigenvaIues and eigenvectors4.5 The finite-dimensional case. Characteristic polynomials If dim V = it, the problem of finding the eigenvalues of a linear transformation T: V+ Vcan be solved with the help of determinants. We wish to find those scalars I such that theequation T(x) = 4x has a solution x with x # 0. The equation T(x) = Ax can be writtenin the form (U- T)(x) = 0,where Zis the identity transformation. If we let TA = AZ - T, then 1 is an eigenvalue if andonly if the equation(4.6) T,(x) = 0has a nonzero solution x, in which case TA is not invertible (because of Theorem 2.10).Therefore, by Theorem 2.20, a nonzero solution of (4.6) exists if and only if the matrix ofT, is singular. If A is a matrix representation for T, then AZ - A is a matrix representationfor TL. By Theorem 3.13, the matrix AZ - A is singular if and only if det (AZ - A) = 0.Thus, if il is an eigenvalue for Tit satisfies the equation(4.7) det (AZ- A) = 0.Conversely, any 1 in the underlying jield of scalars which satisfies (4.7) is an eigenvalue.This suggests that we should study the determinant det (AZ - A) as a function of A. THEOREM 4.4. Zf A is any n x n matrix and lyZ is the n x n identity matrix, thefinctionfdefined by the equation ,f(ii) = det (AZ - A)is a polynomial in 1 of degree n. Moreover, the term of highest degree is I", and the constantterm isf(0) = det (-A) = (- 1)" det A. Proof. The statement f(0) = det (-A) follows at once from the definition of J Weprove thatf is a polynomial of degree n only for the case n 5 3. The proof in the generalcase can be given by induction and is left as an exercise. (See Exercise 9 in Section 4.8.) For n = 1 the determinant is the linear polynomialf(l)= 1 - alI. For n = 2 we have 1 - alI --al2 det (AZ - A) = = (a - ada - a,,) - a12a21 - azl a - a22 = a2 - (all + az2M + (w22 - a12azl>)a quadratic polynomial in 1. For n = 3 we have a - all -al2 -al3det (ill - A) = -a21 I - a22 - a23 -4, -a32 1 - a,,
Calculation of eigenvalues and eigenvectors in thejnite-dimensional case 103 1 - a22 - az3 --a,, - az3 -azl a - az2 = 0 - all> + al2 -a32 1 - a33 I I -a31 1 -. a,, -a31 -a32The last two .terms are linear polynomials in 1. The first term is a cubic polynomial, theterm of highest degree being L3. DEFINITION. If A is an n x n matrix the determinant f(A) = det (AZ - A)is called the characteristic polynomial of A. The roots of the characteristic polynomial of A are complex numbers, some of whichmay be real. If we let F denote either the real field R or the complex field C, we have thefollowing theorem,. THEOREM 43. Let T: V+ V be a linear transformation, where V has scaIars in F, anddim V = n . Let A be a matrix representation of T. Then the set of eigenvalues of T consistsof those roots of the characteristic polynomial of A which lie in F. Proof. The discussion preceding Theorem 4.4 shows that every eigenvalue of T satisfiesthe equation det (AZ - A) = 0 and that any root of the characteristic polynomial of Awhich lies in 17 is an eigenvalue of T. , The matrix .4 depends on the choice of basis for V, but the eigenvalues of T were definedwithout reference t'o a basis. Therefore, the set of roots of the characteristic polynomial ofA must be independent of the choice of basis. More than this is true. In a later section weshall prove that thie characteristic polynomial itself is independent of the choice of basis.We turn now to the problem of actually calculating the eigenvalues and eigenvectors in thefinite-dimensional case.4.6 Calculation of' eigenvalues and eigenvectors in the finite-dimensional case In the finite-dimensional case the eigenvalues and eigenvectors of a linear transformationTare also called eig,envalues and eigenvectors of each matrix representation of T. Thus, theeigenvalues of a square matrix A are the roots of the characteristic polynomial f(A) =det (AZ - A). The eigenvectors corresponding to an eigenvalue A are those nonzerovectorsX= (x.r,. . . , x,) regarded as n x 1 column matrices satisfying the matrix equation AX = IX, or ( A I - A)X= 0 .This is a system of n linear equations for the components x1, . . . , x,. Once we know ;1we can obtain the eigenvectors by solving this system. We illustrate with three examplesthat exhibit difrerent features.
106 Eigenvalues and eigenvectors Xl + x2 + x3 = 0repeated three times. To solve this equation we may take x1 = a, xZ = b , where a and bare arbitrary, and then take x3 = -a - b . Thus every eigenvector corresponding to il = 1has the form (a, b, -a - b) = a(l,O, -1) + b(0, 1, -1),where a # 0 or b # 0. This means that the vectors (1, 0, -1) and (0, 1, -1) form a basisfor E(1). Hence dim E(A) = 2 when il = 1. The results can be summarized as follows: Eigenvalue Eigenvectors dim E(l) 7 t(l,2,3), t#O 1 131 a(1, 0, - 1 ) + b(0, 1, -l), a, b not both 0. 2Note that in this example there are three independent eigenvectors but only two distincteigenvalues.4.7 Trace of a matrix Letf(il) be the characteristic polynomial of an n x n matrix A. We denote the n roots off(4 by 4, . . . 3 A,, with each root written as often as its multiplicity indicates. Then wehave the factorization f(l) = (1 - A,) * * * (A - I,).We can also writef(A) in decreasing powers of I as follows, f(A) = A" + c,-lIz"-l + - * * + Cl2 + co.Comparing this with the factored form we find that the constant term c,, and thecoefficient ofAn-r are given by the formulasSince we also have c,, = (-1)" det A, we see that 1, * * - 1, = det A.That is, theproduct of the roots of the characteristicpolynomialof A is equal to the determinantofA. The sum of the roots off(A) is called the trace of A, denoted by tr A. Thus, by definition, tr A = 5 Ii. i=lThe coefficient of 1*-l is given by c,-r = -tr A. We can also compute this coefficientfrom the determinant form forf(il) and we find that cnel = -(all + es* + ad.
Exercises 107(A proof of this formula is requested in Exercise 12 of Section 4.8.) The two formulas forc,,-~ show that tr A =iaii.That is, the trace of A is also equal to the sum of the diagonal elements of A. Since the sum of the diagonal elements is easy to compute it can be used as a numericalcheck in calculations of eigenvalues. Further properties of the trace are described in thenext set of exerci;ses.4.8 Exercises Determine the eigenvalues and eigenvectors of each of the matrices in Exercises 1 through 3.Also, for eaclh eigenvalue A compute the dimension of the eigenspace E(1). [I 1 a , a > 0, b > 0. 2* b 1 4.ThematricesP,=[T i], Pz=[p -i-J, P,=[:: -T]occurinthequantum mechanical theory of electron spin and are called Pauli spin matrices, in honor of the physicist Wolfgang Pauli (1900-1958). Verify that they all have eigenvalues 1 and - 1 . Then determine all 2 x 2 matrices with complex entries having the two eigenvalues 1 and -1 . 5. Determine all 2 x 2 matrices with real entries whose eigenvalues are (a) real and distinct, (b) real and equal, (c) complex conjugates. 6. Determine a, b, c, d, e, f, given that the vectors (1, 1, l), (1, 0, - l), and (1, - 1 , 0) are eigen- vectors of the matrix 7. Calculate the eigenvalues and eigenvectors of each of the following matrices, Also, compute I the dimension (of the eigenspaceE(A) for each eigenvalue L 1 5 -6 -6 -1 4 2 3 -6 -4 8. Calculate the eligenvalues of each of the five matrices (4 9 (b)
108 Eigenvalues and eigenvectors These are called Dirnc matrices in honor of Paul A. M. Dirac (1902- ), the English physicist. They occur in the solution of the relativistic wave equation in quantum mechanics. 9. If A and B are n x n matrices, with B a diagonal matrix, prove (by induction) that the deter- minant f(n) = det (AB - A) is a polynomial in 1 with f(0) = (-1)" det A, and with the coefficient of I" equal to the product of the diagonal entries of B.10. Prove that a square matrix A and its transpose At have the same characteristic polynomial.11. If A and B are n x n matrices, with A nonsingular, prove that AB and BA have the same set of eigenvalues. Note: It can be shown that AB and BA have the same characteristic poly- nomial, even if A is singular, but you are not required to prove this.12. Let A be an n x n matrix with characteristic polynomialf(l). Prove (by induction) that the coefficient of An-l inf(A) is -tr A.13. Let A and B be II x n matrices with det A = det B and tr A = tr B. Prove that A and B have the same characteristic polynomial if n = 2 but that this need not be the case if n > 2.14. Prove each of the following statements about the trace. (a)tr(A+B)=trA+trB. (b) tr (CA) = c tr A. (c) tr (AB) = tr (BA). (d) tr At = tr A.4.9 Matrices representing the same linear transformation. Similar matrices In this section we prove that two different matrix representations of a linear trans-formation have the same characteristic polynomial. To do this we investigate more closelythe relation between matrices which represent the same transformation. Let us recall how matrix representations are defined. Suppose T: V-t W is a linearmapping of an n-dimensional space V into an m-dimensional space W. Let (e,, . . . , e,)and(w,,..., w,) be ordered bases for V and W respectively. The matrix representation of T relative to this choice of bases is the m x n matrix whose columns consist of the com-ponents of T(e,), . . . ,T(e,) relative to the basis (wl, . . . , w,). Different matrix represen-tations arise from different choices of the bases. We consider now the case in which V = W, and we assume that the same ordered basis h,..., e,) is used for both V and W. Let A = (a+) be the matrix of T relative to this basis. This means that we have(4.8) T(e,) = %tl aikei for k=l,2 ,..., n.Now choose another ordered basis (q , . . . , u,) for both V and Wand let B = (bkj) be thematrix of T relative to this new basis. Then we have
Matrices representing the same linear transformation. Similar matrices 109(4.9) T(ui) = 5 bkju, for j=l,2 ,..., n. k=l Since each ui is in the space spanned by e,, . . . , e, we can write(4.10) uj =$2jek for j = 1,2, . . , n,for some set of scalars cki . The n x n matrix C = (ckJ determined by these scalars is non-singular because it represents a linear transformation which maps a basis of V onto anotherbasis of V. ,4pplying T to both members of (4.10) we also have the equations(4.11) T(uj) = i ckiT(ek) for j=l,2 ,..., n. k=l The systems of equations in (4.8) through (4.11) can be written more simply in matrixform by introducing matrices with vector entries. Let E = [e,,...,e,] and u = [u, , * . . ) u,]be 1 x n row matrices whose entries are the basis elements in question. Then the set ofequations in (4.10) can be written as a single matrix equation,(4.12) U=EC.Similarly, if we introduce E' = [T(e,), . . . , T(e,)] q and U' = [T(q), . . . , T(u,)] , Equations (4.8), (4.9), and (4.11) become, respectively, (4.13) E'=EA, U'= UB, U' = E'C. From (4.12) we also have E = UC-'. To find the relation between A and B we express U' in two ways in terms of U. From (4.13) we have U' = UB and U' = E'C = EAC = UC-IAC, Therefore UB = ZJCYAC. But each entry in this matrix equation is a linear combination
110 Eigenvalues and eigenvectorsof the basis vectors ul, . . . , u,, . Since the ui are independent we must have B = C-'AC.Thus, we have proved the following theorem. THEOREM 4.6. If two n x n matrices A and B represent the same linear transformation T,then there is a nonsingular matrix C such that B = C-lAC.Moreover, if A is the matrix of T relative to a basis E = [el , . , . , e,] and if B is the matrixof T relative to a basis U = [ul, . . . , u,], then for C we can take the nonsingular matrixrelating the two bases according to the matrix equation U = EC. The converse of Theorem 4.6 is also true. THEOREM 4.7. Let A and B be two n x n matrices related by an equation of the formB = C-IAC, where C is a nonsingular n x n matrix. Then A and B represent the samelinear transformation. Proof. Choose a basis E = [e,, . . . , e,] for an n-dimensional space V. Let u,, . . . , u,be the vectors determined by the equations n(4.14) Uj = xckjek for j=l,2 ,..., n, +lwhere the scalars c,, are the entries of C. Since C is nonsingular it represents an invertiblelinear transformation, so U = [uI, . . . , u,,] is also a basis for V, and we have U = EC. Let Tbe the linear transformation having the matrix representation A relative to the basisE, and let S be the transformation having the matrix representation B relative to the basis U.Then we have(4.15) T(ek) = fi aike{ for k = 1,2, . . . , n f=land(4.16) s(U,) = 5 b,,u, for j = 1,2, . . . , n. k==lWe shall prove that S = T by showing that T(u,) = S(u,) for eachj. Equations (4.15) and (4.16) can be written more simply in matrix form as follows,
Matrices representing the same linear transformation. Similar matrices 111 [TM . . . , %Jl = E-4, [SW, - * . , S(u,)] = UB.Applying 7'to (4.14) we also obtain the relation T(z+) = 2 cJ(e,), or [T(q), . . . , T(u,)] = EAC.But we have UB = ECB = EC(C-lAC) = EAC,which shows tha.t T(uJ = S(q) for each j. Therefore T(X) = S(x) for each x in V, so T =S. In other words, the matrices A and B represent the same linear transformation. DEFINITION. 'Two n x n matrices A and B are called similar tf there is a nonsingularmatrix C such that B = C-IAC. Theorems 4.6 and 4.7 can be combined to give us THEOREM ,4.8. Two n x n matrices are similar tfand only tf they represent the same Iineartransformation. Similar matrices share many properties. For example, they have the same determinantsince det (CYAC) = det (C-l)(det A)(det C) = det A.This property gives us the following theorem. T HEOREM 4.9. Similar matrices have the same characteristic polynomial and therefore thesame eigenvalues. Proof. If A and B are similar there is a nonsingular matrix C such that B = C-lAC.Therefore we have ill .- B = LI - C-IAC = ilC-=IC - C-lAC := C-l(ilZ - A)C.This shows that .ill - B and AZ - A are similar, so det (Al- B) = det (AZ - A). Theorems 4.8 and 4.9 together show that all matrix representations of a given lineartransformation T have the same characteristic polynomial. This polynomial is also calledthe characteristic polynomial of T. The next theorem is a combination of Theorems 4.5, 4.2, and 4.6. In Theorem 4.10, Fdenotes either the real field R or the complex field C.
112 Eigenvalues and eigenvectors THEOREM 4.10. Let T: V + V be a linear transformation, where V has scalars in F, anddim V = n. Assume that the characteristic polynomial of T has n distinct roots I,, . . . , 1, inF. Then we have: (a) The corresponding eigenvectors ul, . . . , u, form a basis for V. (b) The matrix of T relative to the ordered basis U = [ul, . . . , u,] is the diagonal matrix A having the eigenvalues as diagonal entries: A=diag(il,,...,il,). (c) If A is the matrix of T relative to another basis E = [e,, . . . , e,], then A = C-lAC, where C is the nonsingular matrix relating the two bases by the equation U=EC. Proof. By Theorem 4.5 each root Ai is an eigenvalue. Since there are n distinct roots,Theorem 4.2 tells us that the corresponding eigenvectors ul, . . . , u, are independent.Hence they form a basis for V. This proves (a). Since T(uJ = &ui the matrix of T relativeto U is the diagonal matrix A, which proves (b). To prove (c) we use Theorem 4.6. Note: The nonsingular matrix C in Theorem 4.10 is called a diagonalizing matrix. If h,..., e,) is the basis of unit coordinate vectors (I,, . . . , I,), then the equation U = EC in Theorem 4.10 shows that the kth column of C consists of the components of the eigenvector ule relative to (Z1, . . . , I,). If the eigenvalues of A are distinct then A is similar to a diagonal matrix. If the eigen-values are not distinct then A still might be similar to a diagonal matrix. This will happenif and only if there are k independent eigenvectors corresponding to each eigenvalue ofmultiplicity k. Examples occur in the next set of exercises.4.10 Exercises Prove that the matrices [i :] and [IO T] have the same eigenvalues but are not similar. In each case find a nonsingular matrix C such that C-IAC is a diagonal matrix or explain why no such C exists. Three bases in the plane are given. With respect to these bases a point has components (x,, x,), (yl,yzh and (q, Z-A respectively. Suppose that [yl, yzl = [xl, x&, h 4 = [xl, x,lB, and [z,, z2] = [yl, y,]C, where A, B, C are 2 x 2 matrices. Express C in terms of A and B.
Exercises 1134. In each case, slhow that the eigenvalues of A are not distinct but that A has three independent eigenvectors. Find a nonsingular matrix C such that C-'AC is a diagonal matrix.5. Show that none of the following matrices is similar to a diagonal matrix, but that each is similar A 0 to a triangular matrix of the form where A is an eigenvalue. [1 I' I ! I 0 -1 06. Determine the eigenvalues and eigenvectors of the matrix 0 0 1 and thereby show that it is not similar to a diagonal matrix. -. 1 -3 37. (a) Prove that a. square matrix A is nonsingular if and only if 0 is not an eigenvalue of A. (b) If A is nonsingular, prove that the eigenvalues of A-l are the reciprocals of the eigenvalues ofA.8. Given an n x n matrix A with real entries such that A2 = -I. Prove the following statements about A. (a) A is nonsingular. (b) n is even. (c) A has no real eigenvalues. (d) det A = 1. q
5 EIGENVALUES OF OPERATORS ACTING ON EUCLIDEAN SPACES5.1 Eigenvalues and inner products This chapter describes some properties of eigenvalues and eigenvectors of linear trans-formations that operate on Euclidean spaces, that is, on linear spaces having an innerproduct. We recall the fundamental properties of inner products. In a real Euclidean space an inner product (x, u) of two elements x andy is a real numbersatisfying the following properties : (1) (X>Y) = 09 4 (symmetry) (2) 6 + z, Y> = (XT Y> + (z9 r> (linearity) (3) (WY) = C(XPY> (homogeneity) (4) (x, 4 > 0 i f x#O (positivity). In a complex Euclidean space the inner product is a complex number satisfying the sameproperties, with the exception that symmetry is replaced by Hermitian symmetry,(1') (x3 JJ> = (y, 4,where the bar denotes the complex conjugate. In (3) the scalar c is complex. From (1')and (3) we obtain(3') (X, Cy) = f(& J,) 3which tells us that scalars are conjugated when taken out of the second factor. Takingx = y in (1') we see that (x, x) is real so property (4) is meaningful if the space is complex. When we use the term Euclidean space without further designation it is to be understoodthat the space can be real or complex. Although most of our applications will be to finite-dimensional spaces, we do not require this restriction at the outset. The first theorem shows that eigenvalues (if they exist) can be expressed in terms of theinner product. THEOREM 5.1. Let E be a Euclidean space, let V be a subspace of E, and let T: V--t E be alinear transformation having an eigenvalue 31 with a corresponding eigenvector x. Then we have(5.1)11A
Hermitian and skew-Hermitian transformations 115 Proof. Since T(x) = iix we have (T(x), x) = (Ax, x) = 1(x, x).Since x # 0 we can divide by (x, x) to get (5.1). Several properties of eigenvalues are easily deduced from Equation (5.1). For example,from the Hermitian symmetry of the inner product we have the companion formula 2 = (x2 T(x))(5.2) (x9 x>for the complex conjugate of A. From (5.1) and (5.2) we see that il is real (1 = 1) if andonly if (T(x), x) is real, that is, if and only if (T(x), 4 = (x, T(x)) for the eigenvector x .(This condition is trivially satisfied in a real Euclidean space.) Also, ;i is pure imaginary(A = -2) if and only if (T( x ) , x 1s p ure imaginary, that is, if and only if ) ' (T(x), 4 = -6~ T(x)) for the eigenvector x.5.2 Hermitian and skew-Hermitian transformations In this section we introduce two important types of linear operators which act on Euclid-ean spaces. These operators have two categories of names, depending on whether theunderlying Euclidean space has a real or complex inner product. In the real case the trans-formations are called symmetric and skew-symmetric. In the complex case they are calledHermitian and skew-Hermitian. These transformations occur in many different applications.For example, Hermitian operators on infinite-dimensional spaces play an important role inquantum mechanics. We shall discuss primarily the complex case since it presents no addeddifficulties. DEFINITION. Let E be a Euclidean space and let V be a sub..l;aace of E. A linear trans-formation T: V--f E is called Hermitian on V if (T(x), y) = (x, T(y)) .for all x and y in V.Operator T is called skew-Hermitian on V if (T(X), y) = - (x, T(y)> .for all x and y in V. In other words, a Hermitian operator Tcan be shifted from one factor of an inner productto the other without changing the value of the product. Shifting a skew-Hermitian operatorchanges the sign of the product. Note: As already mentioned, if Eis a reaLEuclidean space, H'ermitian transformations are also called symmetric; skew-Hermitian transformations are called skew-symmetric.
116 Eigenvalues of operators acting on Euclidean spaces EXAMPLE 1. Symmetry and skew-symmetry in the space C(a, b). Let C(a, b) denote thespace of all real functions continuous on a closed interval [a, b], with the real inner product U-Y s> = jabf(Os(t) dt.Let Vbe a subspace of C(a, b). If T: V -+ C(a, b) is a linear transformation then cf, T(g)) =ji f(t)Tg(t) dt , where we have written Q(t) for T(g)(t). Therefore the conditions for sym-metry and skew-symmetry become(5.3) s," {f(t)Tg(t) - g(t)Tf(t)} dt = 0 if T is symmetric,and(5.4) jab {f(t)Tg(t) + g(t)Tf(t)} dt = 0 if T is skew-symmetric. EXAMPLE 2. Multiplication by a$xedfunction. In the space C(a, b) of Example 1, choosea fixed functionp and define T(f) = pf, the product ofp and f. For this T, Equation (5.3)is satisfied for all f and g in C(a, b) since the integrand is zero. Therefore, multiplicationby a fixed function is a symmetric operator. EXAMPLE 3. The dyerentiation operator. In the space C(a, b) of Example 1, let Y be thesubspace consisting of all functions f which have a continuous derivative in the open interval(a, b) and which also satisfy the boundary condition f (a) = f(b). Let D: V --+ C(a, b) be thedifferentiation operator given by D(f) = f . It is easy to prove that D is skew-symmetric.In this case the integrand in (5.4) is the derivative of the product fg, so the integral is equalto Iab (h)'(t) dt = f@k@) - f(a>g(a) .Since bothfand g satisfy the boundary condition, we have f (b)g(b) - f (a)g(a) = 0. Thus,the boundary condition implies skew-symmetry of D. The only eigenfunctions in the sub-space V are the constant functions. They belong to the eigenvalue 0. EXAMPLE 4. Sturm-Liouville operators. This example is important in the theory of linearsecond-order differential equations. We use the space C(a, b) of Example I once more andlet V be the subspace consisting of all f which have a continuous second derivative in [a, b]and which also satisfy the two boundary conditions(5.5) p(4f (4 = 0 p p(blf(b) = 0,wherep is a fixed function in C(a, b) with a continuous derivative on [a, b]. Let q be anotherfixed function in C(a, b) and let T: V ---f C(a, b) be the operator defined by the equation T(f) = Cpf 7 + qf.
Orthogonality of eigenvectors corresponding to distinct eigenvalues 117This is called a Sturm-Liouville operator. To test for symmetry we note that j?(g) -gT(f) = f(pg')' - dpf') ' Using this in (5.3) and integrating both ji f * (pg')' dt andJ," g * (pf')' dt by parts, we find jab if%) - gT(f)) dt = fm' I;- Iab pg'f' dt - gpf' [I+ 1," pf'g' dt = 0,since bothfand g satisfy the boundary conditions (5.5). Hence T is symmetric on V. Theeigenfunctions of Tare those nonzero f which satisfy, for some real 1, a differential equationof the form Q?fY' + qf = von [a, b], and also satisfy the boundary conditions (5.5).5.3 Eigenvalues and eigenvectors of Hermitian and skew-Hermitian operators Regarding eigenvalues we have the following theorem; THEOREM 5.2. Assume T has an eigenvalue il. Then we have: (a) If T is Hermitian, 1 is real: il = 2. (b) If T is skew-Hermitian, R is pure imaginary: .1 = -1. Proof. Let x be an eigenvector corresponding to 1. Then we have 2 _ (T(x), x> - - and - _ ___- j (x9 T(x)) (x, x> (x,x> *If Tis Hermitian we have (T(x), x) = (x, T(x)) so A = 1. If Tis skew-Hermitian we haveG'W, 4 = -(x, T(x)) so il = -2. Note: If T is symmetric, Theorem 5.2 tells us nothing new about the eigenvalues of T since all the eigenvalues must be real if the inner product is real.. If T is skew-symmetric, the eigenvalues of T must be both real and pure imaginary. Hence all the eigenvalues of a skew-symmetric operator must be zero (if any exist).5.4 Orthogonality of eigenvectors corresponding to distinct eigenvalues Distinct eigenvalues of any linear transformation correspond to independent eigen-vectors (by Theorem 4.2). For Hermitian and skew-Hermitian transformations more istrue. THEOREM 5.3. Let T be a Hermitian or skew-Hermitian transformation, and let L and ,abe distinct eigenvalues of T with corresponding eigenvectors x and y. Then x and y areorthogonal; that is, (x, y) = 0. ProoJ We write T(x) = Lx, T(y) = ,uy and compare the two inner products (T(x), y)and (x, T(y)). We have (T(x), y> = (k y> = W, y> and (x, To)) = (x, luy> = /Xx, y).
118 Eigenvalues of operators acting on Euclidean spacesIf T is Hermitian this gives us 2(x, y) = ,L?(x, y) = ,u(x,~) since ,,v = ,ii. Therefore(x, y) = 0 since il # p. If T is skew-Hermitian we obtain 1(x, y) = -,E(x, y) = ,M(x, y)which again implies (x, J) = 0. EXAMPLE. We apply Theorem 5.3 to those nonzero functions which satisfy a differentialequation of the form(5.6) w-7 + qf= Afon an interval [a, b], and which also satisfy the boundary conditionsp(a)f(a) = p(b)f (b) =0. The conclusion is that ;any two solutions f and g corresponding to two distinct values of;I are orthogonal. For example, consider the differential equation of simple harmonicmotion, f" + k2f= 0 on the interval [0, 7~1, where k + 0. This has the form (5.6) with p = 1, q = 0, and A = -k2. All solutions are given byf(t) = c1 cos kt + c2 sin kt . The boundary conditionf (0) = 0 implies c1 = 0. The second boundary condition, f (7r) = 0, implies cg sin krr = 0. Since c2 # 0 for a nonzero solution, we must have sin kr = 0, which means that k is aninteger. In other words, nonzero solutions which satisfy the boundary conditions exist if and only if k is an integer. These solutions are f(t) = sin nt , n = f 1, i-2, . . . . The orthogonality condition irnplied by Theorem 5.3 now becomes the familiar relation s : sin nt sin mt dt = 0if m2 and n2 are distinct integers.5.5 Exercises 1. Let E be a Euclidean space, let V be a subspace, and let T: V -+ E be a given linear trans- formation. Let ii be a scalar and x a nonzero element of V. Prove that I is an eigenvalue of T with x as an eigenvector if and only if (TW,y) = %y) for every y in E. 2. Let T(x) = cx for every x in a linear space V, where c is a fixed scalar. Prove that T is sym- metric if V is a real Euclidean space. 3. Assume T: I/ -+ V is a Hermitian transformation. (a) Prove that T" is Hermitian for every positive integer n, and that T-l is Hermitian if T is invertible. (b) What can you conclude about Tn and T-l if T is skew-Hermitian? 4. Let T,: V -+ E and T,: V -+ E be two Hermitian transformations. (a) Prove that aT, + bT, is Hermitian for all real scalars a and b. (b) Prove that the product (composition) TIT, is Hermitian if TI and T, commute, that is, if TIT, = T,T,. 5. Let V = Vs(R) with the usual dot product as inner product. Let T be a reflection in the xy- plane; that is, let T(!) = i, T(i) = j, and T(k) = -k. Prove that T is symmetric.
120 Eigenvalues of operators acting on Euclidean spaces interval [ -1 , l] by r(f) = (pf')', wherep(t) = t2 - 1 . The eigenfunction P,(t) belongs to the eigenvalue A = n(n -b 1). In this example the boundary conditions for symmetry are automatically satisfied since p(l) = p( - 1) = 0.5.6 Existence of an orthonormal set of eigenvectors for Hermitian and skew-Hermitian operators acting on finite-dimensional spaces Both Theorems 5.2 and 5.3 are based on the assumption that T has an eigenvalue. Aswe know, eigenvalues need not exist. However, if T acts on ajnite-dimensional complexspace, then eigenvalues always exist since they are the roots of the characteristic polynomial.If T is Hermitian, all the eigenvalues are real. If T is skew-Hermitian, all the eigenvaluesare pure imaginary. We also know that two distinct eigenvalues belong to orthogonal eigenvectors if T isHermitian or skew-Hermitian. Using this property we can prove that T has an orthonormalset of eigenvectors which spans the whole space. (We recall that an orthogonal set is calledorthonorm.al if each of its (elements has norm 1.) THEOREM 5.4. Assume dim V = n and let T: V ---f V be Hermitian or skew-Hermitian.Then there exist n eigenvectors ul, . . . , u, of T which form an orthonormal basis for V.Hence the matrix of T relative to this basis is the diagonal matrix A = diag (1,) . . . , 1,))where lk is the eigenvalue belonging to ule. Proof. We use inductian on the dimension n. If n = 1, then T has exactly one eigen-value. Any eigenvector u1 of norm 1 is an orthonormal basis for V. Now assume the theorem is true for every Euclidean space of dimension n - 1 . Toprove it is also true for V we choose an eigenvalue 1, for T and a corresponding eigenvectoru1 of norm 1. Then T(u,) = &u, and (IuJ = 1 . Let S be the subspace spanned by ul.We shall apply the induction hypothesis to the subspace Sl consisting of all elements in Ywhich are orthogonal to u:~, P = {x ) x E v, (x, 241) = O}.To do this we need to know that dim Sl = n - 1 and that T maps Sl into itself. From Theorem 1.7(a) we know that u1 is part of a basis for V, say the basis(%,VZ,..., vJ. We can assume, without loss in generality, that this is an orthonormalbasis. (If not, we apply the Gram-Schmidt process to convert it into an orthonormalbasis, keeping u1 as the first basis element.) Now take any x in S1 and writeThen x1 = (x, uJ = 0 since the basis is orthonormal, so x is in the space spanned byv2, * . . ) v,. Hence dim SI = n - 1. Next we show that T maps Sl into itself. Assume T is Hermitian. If x E Sl we have (T(x), 4 = 6, W4) = (x, hu,> = Mx., uJ = 0,
Matrix representations for Hermitian and skew-Hermitian operators 121so T(x) E SL . Since T is Hermitian on SL we can apply the induction hypothesis to findthat T has n - 1 eigenvectors u2, . . . , U, which form an orthonormal basis for Sl.Therefore the orthogonal set ur, . . : , U, is an orthonormal basis for V. This proves thetheorem if T is Hermitian. A similar argument works if T is skew-Hermitian.5.7 Matrix representations for Hermitian and skew-Hermitian operators In this section we assume that V is a finite-dimensional Euclidean space. A Hermitianor skew-Hermitian transformation can be characterized in terms of its action on theelements of any basis. THEOREM 5.5. Let (f?,, . . . , e,) be a basis for V and let T: V -+ V be a linear transfor-mation. Then we have: (a) T is Hermitian if and on4 if (T(e)), ei) = (ej, T(ei)) for all i andj. (b) T is skew-Hermitian if and onfy if (T(e,), ei) = - (ej, T(ei))for all i and j. Proof. Take any two elements x and y in V and express each in terms of the basiselements, say x = 1 xjej and y = 2 yiei. Then we haveSimilarly we find (x, T(Y)) = i 2 qiii(q, T(4). +I i=lStatements (a) and (b) following immediately from these equations. Now we characterize these concepts in terms of a matrix representation of T. THEOREM 5.6. Let (e, , . . . , e,) be an orthonormal basis for V, and let A = (aij) be thematrix representation of a linear transformation T: V -+ V relative to this basis. Then wehave: (a) T is Hermitian if and only if aij = riji for all i andj. (b) T is skew-Hermitian if and only ifaij = -Cji for all i andj. Proof. Since A is the matrix of T we have T(eJ = z;==, akjek . Taking the inner productof T(eJ with ei and using the linearity of the inner product we obtainBut (ek , ei) = 0 unless k = i, so the last sum simplifies to aij(ei, ei) = aij since (ei , ei) = 1 .Hence we have aij = (Wj), 4 for all i, j.
122 Eigenvalues of operators acting on Euclidean spacesInterchanging i and j, taking conjugates, and using the Hermitian symmetry of the innerproduct, 'we find aji = (ej, T(4) for all i, j.Now we apply Theorem 5.5 to complete the proof.5.8 Hermitian and skew-Hermitian matrices. The adjoint of a matrix The following definition is suggested by Theorem 5.6. DEFINITION. A square matrix A = (adj) is called Hermitian if aij = cji for all i and j.Matrix A is called skew-hrermitian if aij = -cjji for all i and j. Theorem 5.6 states that a transformation T on a finite-dimensional space V is Hermitianor skew-Hermitian according as its matrix relative to an orthonormal basis is Hermitianor skew-Hermitian. These matrices can be described in another way. Let 2 denote the matrix obtained byreplacing each entry of A by its complex conjugate. Matrix A is called the conjugate of A.Matrix A is Hermitian if and only if it is equal to the transpose of its conjugate, A = At.It is skew-Hermitian if A = -Jt. The transpose of the conjugate is given a special name. DEFINITION OF THE ADJOI'NT OF A MATRIX. For any matrix A, the transpose of the conjugate,xt, is also called the adjoint of A and is denoted by A*. Thus, a square matrix A is Hermitian if A = A*, and skew-Hermitian if A = -A*.A Hermitian matrix is also called self-adjoint. Note: Much of the older matrix literature uses the term adjoint for the transpose of the cofactor matrix, an entirely different object. The definition given here conforms to the current nomenclature in the theory of linear operators.5.9 Diagonalization of a Hermitian or skew-Hermitian matrix THEOREM 5.7. Every n x n Hermitian or skew-Hermitian matrix A is similar to thediagonal matrix A = diag (2,) . . . , I,) of its eigenvalues. Moreover, we have A = C-IAC,where C is a nonsingular matrix whose inverse is its adjoint, C-l = C* . Proof. Let V be the space of n-tuples of complex numbers, and let (e,, . . . , e,) be theorthonormal basis of unit coordinate vectors. If x = 2 xiei and y = zyyiei let the innerproduct be given by (x, y) = 2 XJi . For the given matrix A, let T be the transformationrepresented by A relative to the chosen basis. Then Theorem 5.4 tells us that V has an
Unitary matrices. Orthogonal matrices 123orthonormal basis of eigenvectors (ul, . . . , u,), relative to which T has the diagonalmatrix representation A = diag (1r, . . . , A,), where 1, is the eigenvalue belonging to u,.Since both A! and A represent T they are similar, so we have A = CFAC, where C = (cu)is the nonsingular matrix relating the two bases: [#I 3 * * * 9 u,] = [el , . . . , e,]C.This equation shows that thejth column of C consists of the components of uj relative to(el, . . . , e,). Therefore cij is the ith component of ui. The inner product of uj and Ui isgiven bySince {ur , . . . , u,} is an orthonormal set, this shows that CC* = I, so C-l = C* . Note: The proof of Theorem 5.7 also tells us how to determine the diagonalizing matrix C. We find an orthonormal set of eigenvectors ul, . . . , u, and then use the components of uj (relative to the basis of unit coordinate vectors) as the entries of thejth column of C. [1 2 2 EXAMPLE 1, The real Hermitian matrix A = has eigenvalues il, = 1 and 1, = 6. 2 5The eigenvectors belonging to 1 are t(2, -l), t # 0. Those belonging to 6 are t(l, 2),t # 0. The two eigenvectors u1 = t(2, -1) and u2 = t(l, 2) with t = l/J5 form anorthonormal set. Therefore the matrix 2 1 C=ljT -1 2 [1 [1is a diagonalizing matrix for A. In this case C* = Ct since C is real. It is easily verified 1 0that PAC = 0 6' EXAMPLE 2. If A is already a diagonal matrix, then the diagonalizing matrix C of Theorem5.7 either leaves A unchanged or m&ely rearranges the diagonal entries.5.10 Unitary matrices. Orthogonal matrices DEFINITION. A square matrix A is called unitary if AA* = I. It is called orthogonal ifAAt = I. Note: Every real unitary matrix is orthogonal since A* = At. Theorem 5.7 tells us that a Hermitian or skew-Hermitian matrix can always be diago-nalized by a unitary matrix. A real Hermitian matrix has real eigenvalues and the corre-sponding eigenvectors can be taken real. Therefore a real Hermitian matrix can be
124 Eigenvalues of operators acting on Euclidean spacesdiagonalized by a real orthogonal matrix. This is not true for real skew-Hermitian matrices.(See Exercise 11 in Section 5.11.) We also have the following related concepts. DEFINITION. A square matrix A with real or complex entries is called symmetric if A =At ; it is called skew-symmetric if A = -At. EXAMPLE 3. If A is real, its adjoint is equal to its transpose, A* = At. Thus, every realHermitian matrix is symmetric, but a symmetric matrix need not be Hermitian. 1 -I- i 2, EXAMPLE: 4. If A = thenA=[i+: -ti], At=[Izi 3il] [ 3 -- i 4iI 'andA*=[lii 3_+4:]. EXAMPLE: 5. Both matrices [: :] and [2 y i 2 : i] are Hermitian. The first is sym-metric, the second is not. EXAMPLE 6. Both matrices [i -3 and L ,2] are skew-Hermitian. The first isskew-symmetric, the second is not. EXAMPLE 7. All the diagonal elements of a Hermitian matrix are real. All the diagonalelements of a skew-Hermitian matrix are pure imaginary. All the diagonal elements of askew-symmetric matrix are zero. EXAMPLE 8. For any square matrix A, the matrix B = &(A + A*) is Hermitian, and thematrix C = $(A - A*) is skew-Hermitian. Their sum is A. Thus, every square matrixA can be expressed as a sum A = B + C, where B is Hermitian and C is skew-Hermitian.It is an easy exercise to verify that this decomposition is unique. Also every square matrixA can be expressed uniquely as the sum of a symmetric matrix, &4 + At), and a skew-symmetric matrix, $(A - At). EXAMPLE 9. If A is orthogonal we have 1 = det (AAt) = (det A)(det At) = (det A)2, sodetA= f l .5.11 Exercises 1. Determine which of the following matrices are symmetric, skew-symmetric, Hermitian, skew- Hermitian.
Exercises 125 2. (a) Verify that the 2 x 2 matrix A = is an orthogonal matrix. (b) Let T be the linear transformation with the above matrix A relative to the usual basis {i,i}. Prove that T maps each point in the plane with polar coordinates (r, a) onto the point with polar coordinates (r, w + 0). Thus, T is a rotation of the plane about the origin, 0 being the angle of rotation. 3. Let V be real 3-space with the usual basis vectors i, j, k. Prove that each of the following [1 matrices is orthogonal and represents the transformation indicated. 1 0 0 (a) 0 1 0 (reflection in the xy-plane). 0 0 -1 (reflection through the x-axis). -1 0 0 Cc) o-1 0 (reflection through the origin). [ 0 0 -1 I ( d ) 1: co:0 -s.o] (rotation about the x-axis). 10 sin 0 cos 8 1 -1 0 0 (4 0 cos 19 -sin 19 1 (rotation about x-axis followed by reflection in the yz-plane). 1 0 sin 8 cos e 1 4. A real orthogonal matrix A is called proper if det A = 1 , and improper if det A = -1 . (a) If A is a proper 2 x 2 matrix, prove that A = for some 0. This represents a rotation through an angle 8. (b) Prove that [A 01] and [ -10 (I)] are improper matrices. The first matrix represents a reflection of the xy-plane through the x-axis; the second represents a reflection through the y-axis. Find all improper 2 x 2 matrices. In each of Exercises 5 through 8, find (a) an orthogonal set of eigenvectors for A, and (b)a unitary matrix C such that C-lAC is a diagonal matrix. 9 12 0 -2 5. A = 6. A = [1 [ 12 16 I * [2 0' I 1 3 4 %A= 43 0 1 10.
126 Eigenvalues of operators acting on Euclidean spaces 9. Determine which of the following matrices are unitary, and which are orthogonal (a, b, 0 real).10. The special theory of relativity makes use of the equations x' = a(x -- ut), y'= y, z'= z, t' = a(t - vx/c2), Here u is the velocity o'f a moving object, c the speed of light, and a = c/m. The linear transformation which maps (x, y, z, t) onto (x', y', z', t') is called a Lorentz transformation. (a> Let 63, ~2, x3, xq) q = (x, y, z, ict) and (xi, xi, xi, xi) = (x', y', z', ict'). Show that the four equations can be written as one matrix equation, a 0 0 -iav/c 0 1 0 0 b;,x;;,x;, x;1 = [Xl, x2,x,, xql i 0 0 1 0 1 * liavjc 0 0 a J [ I. (b) Prove that the 4 x 4 matrix in (a) is orthogonal but not unitary. 0 a11. Let a be a nonzero real number and let A be the skew-symmetric matrix A = --a 0 (a) Find an orthonormal set of eigenvectors for A. (b) Find a unitary matrix C such that @AC is a diagonal matrix. (c) Prove that there is no real orthogonal matrix C such that CplAC is a diagonal matrix.12. If the eigenvalues of a Hermitian or skew-Hermitian matrix A are all equal to c, prove that A =cl.13. If A is a real skew-symmetric matrix, prove that both Z - A and Z + A are nonsingular and that (I - A)(Z + A)-1 is orthogonal.14. For each of the following statements about n x n matrices, give a proof or exhibit a counter example. (a) If A and B are unitary, then A + B is unitary. (b) If A and B are unitary, then AB is unitary. (c) If A and AB are unitary, then B is unitary. (d) If A and B are unitary, then A + B is not unitary.5.12 Quadratic forms Let V be a real Euclidea:n space and let T: V+ V be a symmetric operator. This meansthat T can be shifted from one factor of an inner product to the other, V(x), y) = (x, T(y)) forallxandyin V.Given T, we define a real-valued function Q on V by the equation Q(x) = UW 4.
Quadratic forms 127The function Q is called the quadratic form associated with T. The term "quadratic" issuggested by the following theorem which shows that in the finite-dimensional case Q(x)is a quadratic polynomial in the components of x. THEOREM 5.8. Let (e,, . . . , e,) be an orthonormal basis for a real Euclidean space V.Let T: V + V be a symmetric transformation, and let A = (aii) be the matrix of T relativeto this basis. Then the quadratic form Q(x) = (T(x), x) is related to A as follows:(5.7) Q(X) =i~~~i+ixi !f x =i2xieie Proof. By linearity we have T(x) = 2 xiT(eJ . ThereforeThis proves (5.7) since aij = aji = (T(ei), ej). The sum appearing in (5.7) is meaningful even if the matrix A is not symmetric. DEFINITION. Let V be any real Euclidean space with an orthonormal basis (e, , . . . , e,), andlet A = (aij) by any n x n matrix of scalars. The scalar-valued function Q defined at eachelement x = 2 xiei of V by the double sum(5.8) Q(X) = iz $FilXiXiis called the quadratic form associated with A. If A is a diagonal matrix, then aij = 0 if i # j so the sum in (5.8) contains only squaredterms and can be written more simply as Q(x) = i$l w,2 .In this case the quadratic form is called a diagonalform. The double sum appearing in (5.8) can also be expressed as a product of three matrices. THEOREM 5.9. Let x = [Xl,.. . , x,] be a 1 x n row matrix, and let A = (a,J be ann x n matrix. Then XAXt is a 1 x 1 matrix with entry(5.9) Proof. Th.e product XA is a 1 x II matrix, XA = [yl, . . . , y,] , where entry yj is the dotproduct of X with thejth column of A, yi = i xiaij. i=l
128 Eigenvalues of operators acting on Euclidean spacesTherefore the product XLF is a 1 x 1 matrix whose single entry is the dot product Note: It is customary to identify the 1 x 1 matrix XAXt with the sum in (5.9) and to call the product XAXt a. quadratic form. Equation (5.8) is written more simply as follows: Q(x) = XAXt . [ 1 1 - 1 EXAMPLE 1. Let zi = X = [x1, x2]. Then we have 1 .- 3 5' [ 1 - 1 XA = I'X,) x2] = [Xl - 3x,, -x1 + 5x,1, - 3 5and hence XAXt = [Xl -- 3x,, -x1 + 5x,] = x; - 3x,x, - XlX'J + 5x;. EXAMPLE 2. LetB = X = [x1, x2]. Then we have XBXt = [xl,,x2][-; -3[::1 = x; -2x,x, -2x,x, + 5x2,. In both Examples 1 and 2 the two mixed product terms add up to -4x1x, so XAF =X&V. These examples show that different matrices can lead to the same quadratic form.Note that one of these matrices is symmetric. This illustrates the next theorem. THEOREM 5.10. For any n x n matrix A and any 1 x n row matrix X we have XAXt =XBXt where B is the symmetric matrix B = +(A + At). Proof. Since XAXt is a 1 x 1 matrix it is equal to its transpose, XAXt = (XAXt)t.But the transpose of a product is the product of transposes in reversed order, so we have(XAXt)t = XAtXt . Therefore XAXt = &XAXt + *XAtXt = XBXt .5.13 Reduction of a real quadratic form to a diagonal form A real symmetric matrix A is Hermitian. Therefore, by Theorem 5.7 it is similar to thediagonal matrix A = diag (jll, . . . , A,) of its eigenvalues. Moreover, we have A = CtAC,where C is an orthogonal matrix. Now we show that C can be used to convert the quadraticform XAXt to a diagonal form.
Reduction of a real quadratic form to a diagonal form 129 THEOREM 5.11. Let XAF be the quadratic form associated with a real symmetric matrixA, and let C be an orthogonal matrix that converts A to a diagonal matrix A = FAC.Then we have XAXt = YAYt = i&y;, i=lwhere Y = [;vI, . ,I . , y,] is the row matrix Y = XC, and AI, . . . ,I,, are the eigenvalues of A. Proof. Since C is orthogonal we have C-l = Ct. Therefore the equation Y = XCimplies X = YCt , and we obtain XAXt = ( YCt)A( YCt)t = Y(CtAC) Yf = YA Yt . Note: Theorem 5.11 is described by saying that the linear transformation Y = XC reduces the quadratic form X,4X1 to a diagonal form YA Yt. EXAMPLE 1. The quadratic form belonging to the identity matrix is XI&v =tglxf = I(Xl12,the square of the length of the vector X = (x1, . , . , x,) . A linear transformation Y = XC,where C is an orthogonal matrix, gives a new quadratic form YAYt with A = CZCt =CC? = I. Since XZXt = YZYt we have I(X(12 = I( Y112, so Y has the same length as X. Alinear transformation which preserves the length of each vector is called an isometry.These transformations are discussed in more detail in Section 5.19. EXAMPLE 2.. Determine an orthogonal matrix C which reduces the quadratic form Q(x) =2x,2 + 4x,x, + 5x-i to a diagonal form. [1 2 2 Solution. We write Q(x) = XAXt, where A = This symmetric matrix was 2 5.diagonalized in Example 1 following Theorem 5.7. It has the eigenvalues 1, = 1, AZ = 6,and an orthonormal set of eigenvectors ul, u2, where u1 = t(2, -l), u2 = t(l,2), t = [1 2 1l/J?. An orthogonal diagonalizing matrix is C = t The correspondingdiagonal form is -1 2 . YAYt = il,y; + I,y; = y; + 6-y;. The result of Example 2 has a simple geometric interpretation, illustrated in Figure 5.1.The linear transformation Y =: XC can be regarded as a rotation which maps the basis i,j onto the new basis ul, u2. A point with coordinates (x1, xJ relative to the first basis has new coordinates (yl, yJ relative to the second basis. Since XAF = YAYt , the set of points (x1, x2) satisfying the equation XAXt = c for some c is identical with the set of points (yl, y2) satisfying YA Yt = c . The second equation, written as y: + 6~: = c, is
130 Eigenvalues of operators acting on Euclidean spaces to basis i,j to basis u,, FIGURE 5.1 Rotation of axes by an orthogonal matrix. The ellipse has Cartesian equation XAXt = 9 in the jc,x,-system, and equation YA Yt = 9 in the y&-system.the Cartesian equation of an ellipse if c > 0. Therefore the equation XL%? = c, writtenas 2x: + 4x,x, + 5x: = c, represents the same ellipse in the original coordinate system.Figure 5.1 shows the ellipse corresponding to c = 9.5.14 Applications to analytic geometry The reduction of a quadratic form to a diagonal form can be used to identify the set ofall points (x, v) in the pla.ne which satisfy a Cartesian equation of the form(5.10) ax2 + bxy + cy2 + dx + ey +f= 0.We shall find that this set is always a conic section, that is, an ellipse, hyperbola, parabola,or one of the degenerate cases (the empty set, a single point, or one or two straight lines).The type of conic is governed by the second-degree terms, that is, by the quadratic formax2 + bxy + cy2. To conform with the notation used earlier, we write x1 for X, x2 for y,and express this quadratic form as a matrix product, XAXt = ax,2 + bx,x2 + cxi,where X = [x1, x2] and 11 = By a rotation Y = XC we reduce this form toa diagonal form L,y,2 + 3c2yi, where 1,) A2 are the eigenvalues of A. An orthonormal setof eigenvectors ul, u2 determines a new set of coordinate axes, relative to which the Cartesianequation (5.10) becomes(5.11) 4Yf + i& + d'y, + ey, + f = 0,with new coefficients d' and e' in the linear terms. In this equation there is no mixed productterm yly2, so the type of conic is easily identified by examining the eigenvalues 1, and il, .
Applications to analytic geometry 131If the conic is not degenerate, Equation (5.11) represents an ellipse if 1,) 1, have the samesign, a hyperbola if &, 1, have opposite signs, and a parabola if either 31, or ;1, is zero.. Thethree cases correspond to ;Z,;i., > 0, I,& < 0, and J,J, = 0. We illustrate with somespecific examples. EXAMPLE 1. 2x" + 4xy + 5y2 + 4x + 13y - a = 0. We rewrite this as(5.12) 2~,2+4x~x~+5x~+4x~+13x~-~=O.The quadratic form 2x; -l- 4x,.x, + 5x: is the one treated in Example 2 of the foregoingsection. Its matrix has eigenvalues 1, = 1 , 1, = 6, and an orthonormal set of eigenvectorsu1 = t(2, - I), u2 = t(l,2), where t = l/45. An orthogonal diagonalizing matrix is 2 1C=t This reduces the quadratic part of (5.12) to the form yf + 6~:. To [ -1 2 I *determine the effect on the linear part we write the equation of rotation Y = XC in theform X = YCt and obtain [Xl? x21 = 1 Lh ? [ J5 Y21 2 1 - 1 2 I 9 x1= -@1+ Y2), x2 = +1 + 2Y2).Therefore th.e linear part 4x, + 13x, is transformed to $(2. + Y2) + $(-,I + 2~2) = -$y, + 643~~~.The transformed Cartesian equation becomesBy completing the squares in y1 and y, we rewrite this as follows: (yl - bh2 + 6(y2 + $&)2 = 9.This is the (equation of an ellipse with its center at the point (f&, -&) in the y1y2-system. The positive directions of the yr and y2 axes are determined by the eigenvectors urand u2, as indicated in Figure 5.2. We can simplify the equation further by writing I1==y1-&5, z2=y2+4J5.Geometrically, this is the same as introducing a new system of coordinate axes parallel tothe yly2 axes but with the new origin at the center of the ellipse. In the z,=,-system theequation of the ellipse is simply 2:: + 6z", = 9, orThe ellipse and all three coordinate systems are shown in Figure 5.2.
Eigenvalues of a symmetric transformation obtained as values of its quadratic form 135k 5.16t Eigenvalues of a symmetric transformation obtained as values of its quadratic form Now we drop the requirement that V be finite-dimensional and we find a relation between the eigenvalues of a symmetric operator and its quadratic form. Suppose x is an eigenvector with norm 1 belonging to an eigenvalue A. Then T(x) = Ax so we have (5.13) Q(x) = (T(x), x) = (Ax, x) = A(x, x) = 2, since (x, x) =I 1. The set of all x in V satisfying (x, x) = 1 is called the unit sphere in V. Equation (5.13) proves the following theorem. THEOREM 5.12. Let T: V + V be a symmetric transformation on a real Euclidean space V, and let Q(x) = (T(x), x) . Then the eigenvalues of T (ifany exist) are to be found among the q values that Q takes on the unit sphere in V. L,et V = V,(R) with the usual basis (i,j) and the usual dot product as inner [1 EXAMPLE. 4 0 product. Let T be the symmetric transformation with matrix A = Then the quadratic form of T is given by 0 8' Q(X) = i iaijxixj = 4x,2 + 8x:. i=l j=l The eigenvalues of,T are il, = 4, 1, = 8. It is easy to see that these eigenvalues are, respectively, the minimum and maximum values which Q takes on the unit circle xf + xi = 1 . In fact, on this circle we have Q(x) = 4(x; + x;) + 4x; = 4 + 4x;, where - 1 IX,&. This has its smallest value, 4, when x2 = 0 and its largest value, 8, when x2 = f 1 . Figure 5.4 shows the unit circle and two ellipses. The inner ellipse has the Cartesian equation 4~; -+ 8x: = 4. It consists of all points x = (x1, x,) in the plane satisfying Q(x) = 4. The outer ellipse has Cartesian equation 4x,2 + 8x: = 8 and consists of all points satisfying Q(x) = 8. The points (* I, 0) where the inner ellipse touches the unit circle are eigenvectors belonging to the eigenvalue 4. The points (0, fl) on the outer ellipse are eigenvectors belonging to the eigenvalue 8. The foregoing example illustrates extremal properties of eigenvalues which hold more generally. In the next section we will prove that the smallest and largest eigenvalues (if they exist) are always the minimum and maximum values which Q takes on the unit sphere. Our discussion of these extremal properties will make use of the following theorem on quadratic forms. It should be noted that this theorem does not require that V be finite dimensional. t Starred sections can be omitted or postponed without loss in continuity.
136 Eigenvalues of operators acting on Euclidean spaces - X2 Eigenvector belonging to 8 t / Q(x) = 8 on this ellipse t Xl Eigenvector belonging to 4 Q(x) = 4 on this ellipse FIG~JRE 5.4 Geometric relation between the eigenvalues of T and the values of Q on the unit sphere, illustrated with a two-dimensional example. THEORE:M 5.13. Let T: V + V be a symmetric transformation on a real Euclidean space V with quadratic form Q(x) = (T(x), x) . Assume that Q does not change sign on V. Then if Q(x) = 0 for some x in V we also have T(x) = 0. In other words, ifQ does not change sign, then Q vanishes only on the null space of T. Proof. Assume Q(x) = 0 for some x in V and let y be any element in V. Choose any real t and consider Q(x + ty). Using linearity of T, linearity of the inner product, and symmetry of T, we have Q<x + 9') = G-(x + ty), x + ty) = (T(x) + tUy), x + ty) = (T(x), 4 + V(x), y) + G-(y), 4 + t"(TCy), y) = Q(x) + 2t(T(x), y) + f2Q(y> = at + bt2, where a = 2(T(x), y) and b = Q(y). If Q is nonnegative on V we have the inequality at + bt2 > 0 for all real t . In other words, the quadratic polynomial p(t) = at + bt2 has its minimum at t = 0. Hence p'(0) = 0. But p'(0) = a = 2(T(x), y), so (T(x), y) = 0. Since y was arbitrary, we can in particular take y = T(x), getting (T(x), T(x)) = 0. This proves that T(x) = 0. If Q is nonpositive on V we get p(t) = at + bt2 < 0 for all t, sop has its maximum at t = 0, and hencep'(0) =: 0 as before.* 5.17 Extremal properties of eigenvalues of a symmetric transformation Now we shall prove that the extreme values of a quadratic form on the unit sphere are eigenvalues'. THEOREM 5.14. Let T: V--f V be a symmetric linear transformation on a real Euclidean space V, and let Q(x) = (T(x), x) . Among all values that Q takes on the unit sphere, assume
The finite-dimensional case 137 there is an exlremumt (maximum or minimum) at apoint u with (u, u) = 1 . Then uisaneigen- vector for T; the corresponding eigenvalue is Q(u), the extreme value of Q on the unit sphere. Proof. Assume Q has a minimum at u. Then we have (5.14) Q(x) 2 Q(u) for all x with (x, x) = 1. Let jz = Q(u). If (x, x) = 1 we have Q(u) = 1(x, x) = (lx, x) so inequality (5.14) can be written as (5.15) (T(x), xk 2 @x, 4 provided (x, ,x) = 1. Now we prove that (5.15) is valid for all x in V. Suppose llxll = a. Then x = ay , where llyll = 1. Hence (T(x), x> = (T@yh a.v> = a2(TCy),y) and (k 4 = a"<+, y>. But (TCy), y> 2 (iiy, y) since (y, y> = 1. Multiplying both members of this inequality by a2 we get (5.15) for x = ay . Since (T(x), x) - (Ax, x) = (T(x) - 1 x, x) , we can rewrite inequality (5.15) in the form (T(x) - lx, x) 2 0, or (5.16) (S(x), 4 2 0, where S = T - 2. When x = u 'we have equality in (5.14) and hence also in (5.16). The linear transformation S is symmetric. Inequality (5.16) states that the quadratic form Q1 given by Q,(x) = (S(x), x) is nonnegative on V. When x = u we have Q,(u) = 0. Therefore, by Theorem 5.13 we must have S(U) = 0. In other words, T(u) = iiu, so u is an eigenvector for T, and 1 = Q(u) is the corresponding eigenvalue. This completes the proof if Q has a minimum at u. If there is a maximum at u all the inequalities in the foregoing proof are reversed and we apply Theorem 5.13 to the nonpositive quadratic form Q, .t 5.18 The finitedimensional case Suppose now that dim V = n . Then T has n real eigenvalues which can be arranged in increasing order, say n,<?b,<***<I,. According to Theorem 5.14, the smallest eigenvalue L, is the minimum of Q on the unit sphere, and the largest eigenvalue is the maximum of Q on the unit sphere. Now we shall show that thle intermediate eigenvalues also occur as extreme values of Q, restricted to certain subset.s of the unit sphere. t If Vis infinite-dimensional, the quadratic form Q need not have an extremum on the unit sphere. This will be the case when T has no eigenvalues. In the finite-dimensional case, Q always has a maximum and a minimum somewhere on the unit sphere. This follows as a consequence of a more general theorem on extreme values of continuous functions. For a special case of this theorem see Section 9.16.
138 Eigenvalues of operators acting on Euclidean spaces~_ Let u1 be an eigenvector on the unit sphere which minimizes Q. Then 1, = Q(uJ . If ilis an eigenvalue different from 1, any eigenvector belonging to 1 must be orthogonal to ul.Therefore it is natural to search for such an eigenvector on the orthogonal complement ofthe subspace spanned by ul. Let S be the subspace spanned by ul. The orthogonal complement S'- consists of allelements in V orthogonal to ul. In particular, S'- contains all eigenvectors belonging toeigenvalues il # 1,. It is easily verified that dim SI = n - 1 and that T maps Sl intoitself.? Let Snwl denote the unit sphere in the (n - l)-dimensional subspace S'. (Theunit sphere S,-, is a subset of the unit sphere in V.) Applying Theorem 5.14 to the subspaceSl we find that 1, = Q(uJ , where u2 is a point which minimizes Q on S,-l. The next eigenvector il, can be obtained in a similar way as the minimum value of Q onthe unit sphere S,-Z in the (n - 2)-dimensiontil space consisting of those elements orthog-onal to both u1 and u2. Continuing in this manner we find that each eigenvalue & is theminimum value which Q takes on a unit sphere Sn--k+l in a subspace of dimension n - k + 1.The largest of these minima, A,, is also the maximum value which Q takes on each of thespheres Sn--k+l. The corresponding set of eigenvectors ul, . . . , u, form an orthonormalbasis for V.5.19 Unitary transformations We conclude this chapter with a brief discussion of another important class of trans-formations known as unitary transformations. In the finite-dimensional case they arerepresented by unitary matrices. DEFINITION. Let E be a Euclidean space and V a subspace of E. A linear transformationT: V + E is called unitary on V if we have(5.17) (T(x), T(y)) = (x, y) for all x and y in V.When E is a real Euclidean space a unitary transformation is also called an orthogonaltransformation. Equation (5.17) is described by saying that T preserves inner products. Therefore it isnatural to expect that T also preserves orthogonality and norms, since these are derivedfrom the inner product. THEOREM 5.15. IfT: V + E is a unitary transformation on V, then for all x and y in V wehave: (a) (x, y) = 0 implies (T(x), T(y)) = 0 (Tpreserves orthogonality). @I IIW)ll = llxll (Tpreserves norms). Cc> IIT(x) - T(y)/I = llx - yll (Tpreserves distances). . (d) T is invertible, and T-l is unitary on T(V).t This was done in the proof of Theorem 5.4, Section 5.6.
Unitary transformations 139 Proof. Part (a) follows at once from Equation (5.17). Part (b) follows by taking x = yin (5.17). Part (c) follows from (b) because T(x) - T(y) = T(x - y). To prove (d) we use (b) which shows that T(x) = 0 implies x = 0, so T is invertible.If x E T(V) and y E: T(V) we can write x = T(u), y = T(v), so we have (T-W> Wy>> = (u, u) = (T(u), T(v)) = (x,y).Therefore T-l is unitary on T(V). Regarding eigenvalues and eigenvectors we have the following theorem. THEOREM 5.116. Let T: V -+ E' be a unitary transformation on V. (a) Zf T has an eigenvalue I, then 11) = 1 . (b) Zf x and y are eigenvectors belonging to distinct eigenvalues 1 and ~1, then x and y are orthogonal. (c) Zf V = E and dim V = n, and if V is a complex space, then there exist eigenvectors Ul,... , u, cLf T which form an orthonormal basis for V. The matrix of T relative to this basis is the diagonal matrix A = diag (Al, . . . , A,), u>here 1, is the eigenvalue belonging to uk . Proof. To prove (a), let x be an eigenvector belonging to il. Then x # 0 and T(x) = 31x.Taking y = x in Equation (5.17) we get (lx, Ax) = (x, x) or nqx, x) = (x, x).Since (x, x) > 0 and Ai = 1112, this implies 111 = 1. To prove (b), write T(x) = Lx, T(y) = ,uy and compute the inner product (T(x), T(y))in two ways. We have (T(x), T(Y)) = CGY)since T is unitary. We also have ( WA T(y)> = Gk py) = 44x, y)since x and y are eigenvectors. Therefore 1,6(x, y) = (x, y) , so (x, y) = 0 unless l/i = 1 .But AX = 1 by (a), so if we had A/i = 1 we would also have 21 = Ap ,x = p , A = ,u, whichcontradicts the assumption that A and ,U are distinct. Therefore Ap # 1 and (x, y) = 0. Part (c) is proved by induction on n in much the same way that we proved Theorem 5.4,the corresponding result for Hermitian operators. The only change required is in that partof the proof which shows that T maps SL into itself, where P-={xIxEV, (x,u1)=0}.Here u1 is an eigenvector of T with eigenvalue &. From the equation T(u,) = i,u, we find Ul = a;T(u,) = XIT(ul)
140 Eigenvalues of operators acting on Euclidean spacessince A,& = 11,12 = 1 . Now choose any x in SL and note thatHence T(x) E SL if x E SL, so T maps S' into itself. The rest of the proof is identical withthat of Theorem 5.4, so we shall not repeat the details. The next two theorems describe properties of unitary transformations on a finite-dimensional space. We give only a brief outline of the proofs. THEOKEM 5.17. Assume dim V = n and let E = (e,, . . . , e,) be a jixed basis for V.Then a linear transformation T: V + V is unitary if and only if(5.18) (T(eJ, T(e,)) = (ei , ej) for all i and j .In particular, ifE is orthonormal then T is unitary ifand only ifT maps E onto an orthonormalbasis. Sketch of proof. Write x = 2 xiei , y = c yjej . Then we haveNow compare (x, y) with (T(x), T(y)). THEOREM 5.18. Assume dim V = n and let (e, , . . . , e,) be an orthonormal basis for V.Let A = (aij) be the matrix representation of a linear transformation T: V + V relative tothis basis. Then T is unitary if and only ifA is unitary, that is, if and only if(5.19) A*A = I. Sketch of proof. Since (ei, eJ is the g-entry of the identity matrix, Equation (5.19)implies(5.20)Since A is the matrix of T we have T(eJ = x;=, a,,e,, T(eJ = cFzt=, a,+e,, soNow compare this with (5.20) and use Theorem 5.17.
Exercises 141 THEOREM 5.19. Every unitary matrix A has the following properties: (a) A is nonsingular and A-l = A*. (b) Each of At, /i,and A* is a unitary matrix. (c) The eigenvalues of A are complex numbers of absolute value 1. (d) ldet Al == 1; if A is real, then det A = f 1 . The proof oif Theorem 5.19 is left as an exercise for the reader.5.20 Exercises 1. (a) Let T: I' --t I' be the transformation given by T(x) = cx, where c is a fixed scalar. Prove that T is unitary if and only if ICI = 1 . (b) If V is one-dimensional, prove that the only unitary transformations on V are those des- cribed in (a). In particular, if I/is a real one-dimensional space, there are only two orthogonal transformanons, T(x) = x and T(x) = --x. 2. Prove each of the following statements about a real orthogonal n x n matrix A. (a) If 1 is a real eigenvalue of A, then 1 = 1 or 1 = - 1 . (b) If I is a complex eigenvalue of A, then the complex conjugate 1 is also an eigenvalue of A. In other words, the nonreal eigenvalues of A occur in conjugate pairs. (c) If n is odd, then A has at least one real eigenvalue. 3. Let V be a real Euclidean space of dimension n. An orthogonal transformation T: V + V with determinant 1 is called a rotation. If n is odd, prove that 1 is an eigenvalue for T. This shows that every rotation of an odd-dimensional space has a fixed axis. [Hint: Use Exercise 2.1 4. Given a real orthogonal matrix A with -1 as an eigenvalue of multiplicity k. Prove that det A = (-1)". 5. If T is linear and norm-preserving, prove that T is unitary. 6. If T: V - V is both unitary and Hermitian, prove that T2 = I. 7. Let (e,, . . . , e,) and (u,, . . . , u,) be two orthonormal bases for a Euclidean space V. Prove that there is a unitary transformation T which maps one of these bases onto the other. 8. Find a real (I such that the following matrix is unitary: 9. If A is a skew-Hermitian matrix, prove that both Z - A and Z + A are nonsingular and that (Z - A)(Z + A)-l is unitary.10. If A is a unitary matrix and if Z + A is nonsingular, prove that (I - A)(Z + A)-l is skew- Hermitian.11. If A is Hermitian, prove that A - iZ is nonsingular and that (A - iZ)-l(A + U) is unitary.12. Prove that any unitary matrix can be diagonalized by a unitary matrix.13. A square ma.trix is called normal if AA* = A*A . Determine which of the following types of matrices are normal. (a) Hermitian matrices. (d) Skew-symmetric matrices. (b) Skew-Hermitian matrices. (e) Unitary matrices. (c) Symmetric matrices. (f) Orthogonal matrices.14. If A is a normal matrix (AA* = A*A) and if Cr is a unitary matrix, prove that U*AU is normal.
6 LINEAR DIFFERENTIAL EQUATIONS6.1 Historical introduction The history of differential equations began in the 17th century when Newton, Leibniz, andthe Bernoullis solved some simple differential equations of the first and second order arisingfrom problems in geometry and mechanics. These early discoveries, beginning about 1690,seemed to suggest that the solutions of all differential equations based on geometric andphysical problems could be expressed in terms of the familiar elementary functions ofcalculus. Therefore, much of the early work was aimed at developing ingenious techniquesfor solving differential equations by elementary means, that is to say, by addition, sub-traction, multiplication, division, composition, and integration, applied only a finitenumber of times to the familiar functions of calculus. Special methods such as separation of variables and the use of integrating factors weredevised more or less haphazardly before the end of the 17th century. During the 18thcentury, more systematic procedures were developed, primarily by Euler, Lagrange, andLaplace. It soon became apparent that relatively few differential equations could be solvedby elementary means. Little by little, mathematicians began to realize that it was hopelessto try to discover methods for solving all differential equations. Instead, they found it morefruitful to ask whether or not a given differential equation has any solution at all and,when it has, to try to deduce properties of the solution from the differential equation itself.Within this framework, mathematicians began to think of differential equations as newsources of functions. An important phase in the theory developed early in the 19th century, paralleling thegeneral trend toward a more rigorous approach to the calculus. In the 1820's, Cauchyobtained the first "existence theorem" for differential equations. He proved that everyfirst-order equation of the form Y' = fc? .v)has a solution whenever the right member, S(x,r), satisfies certain general conditions.One important example is the Ricatti equation y' = P(x>y" + QWy + W-4,where P, Q, and R are given functions. Cauchy's work implies the existence of a solutionof the Ricatti equation in any open interval (-r, r) about the origin, provided P, Q, and142
Review of results concerning linear equations of first and second orders 143R have power-series expansions in (-r, r). In 1841 Joseph Liouville (1809-1882) showedthat in some cases this solution cannot be obtained by elementary means. Experience has shown that it is difficult to obtain results of much generality aboutsolutions of differential equations, except for a few types. Among these are the so-calledlinear differential equations which occur in a great variety of scientific problems. Somesimple types were discussed in Volume I-linear equations of first order and linear equationsof second order with constant coefficients. The next section gives a review of the principalresults concerning these equations.6.2 Review of results concerning linear equations of first and second orders A linear differential equation of first order is one of the form(6.1) Y' + P(x>Y = Q(x) 7where P and Q are given functions. In Volume I we proved an existence-uniquenesstheorem for this equation (Theorem 8.3) which we restate here. THEOREM 6.1. Assume P and Q are continuous on an open interval J. Choose any point ain J and let b be any real number. Then there is one and on4 one function y = f (x) whichsatisfies the difSere)rltiaI equation (6.1) and the initial condition f(a) = b . This function isgiven by the explicit formula f(x) = be-A(") + e-dd(6.2) saz Q(t)tict' dt ,where A(x) = jz P(t) dt . Linear equations of second order are those of the form P,,(x)y" + P&)y' + P,(x)y = R(x).If the coefficients PO, PI, P, and the right-hand member R are continuous on some intervalJ, and if P,, is never zero on J, an existence theorem (discussed in Section 6.5) guaranteesthat solutions always exist over the interval J. Nevertheless, there is no general formulaanalogous to (6.2) for expressing these solutions in terms of PO, P, , P,, and R. Thus, evenin this relatively simple generalization of (6.1), the theory is far from complete, except inspecial cases. If the coefficients are constants and if R is zero, all the solutions can bedetermined explicitly in terms of polynomials, exponential and trigonometric functions bythe following theorem which was proved in Volume I (Theorem 8.7). THEOREM 6.2. Consider the dyerential equation(6.3) y"+ay'+by=O,
144 Linear differential equations~~where a and b are given real constants. Let d = a2 - 4b. Then every solution of (6.3) on theinterval (- co, + 00) has the form(6.4) y = e-~r'2[c1u1(x) + c2u2(x)l,where c1 and c2 are constants, and the functions u1 and u2 are determined according to thealgebraic sign of d as follows: (a) If d = 0, then ul(x) = 1 and u2(x) = x . (b) If d > 0, then ul(x) = elcr and u2(x) = eWkx, where k = &6. _ (c) If d < 0, then ul(x) = cos kx and u2(x) = sin kx, where k = &f-d. The number d = a2 - 4b is the discriminant of the quadratic equation(6.5) r2+ar+b=0.This is called the characteristic equation of the differential equation (6.3). Its roots are givenbY - a + Jd - a - Jcl rl = 2 , r2= 2 .The algebraic sign of d determines the nature of these roots. If d > 0 both roots are realand the solution in (6.4) can be expressed in the form y = cleT1" + c2erzx.If d < 0, the roots rl and r2 are conjugate complex numbers. Each of the complexexponential functions fi(x) = er12 and fJ,x) = e7zz is a complex solution of the differentialequation (6.3). We obtain real solutions by examining the real and imaginary parts offiand f2. Writing rl = -ia + ik , r2 = -&a - ik, where k = &J-d, we have fi(x) =: erlx = e-ax12eikz = e-azI2cos kx + ie-arl'sin kxand f2(x) = e'?zx = e--nzfze--iks = e-a"'2cos kx _ ie-a"'"sin kx.The general solution appearing in Equation (6.4) is a linear combination of the real andimaginary parts of fi(x) and f2(x).6.3 Exercises These exercises have been selected from Chapter 8 in Volume I and are intended as a review ofthe introductory material on linear differential equations of first and second orders. Linear equations offirst order. In Exercises 1,2,3, solve the initial-value problem on the specifiedinterval. 1.y'-3y=e2Zon(-C0, +co),withy = O w h e n x = O . 2.xy'-2y=x50n(0,+oo),withy=1whenx=1. 3.y'+ytanx=sin2xon(-6r,$a),withy=2whenx=O.
Linear deferential equations of order n 145 4. If a strain of bacteria grows at a rate proportional to the amount present and if the population doubles in one hour, by how much will it increase at the end of two hours? 5. A curve with Cartesian equation y =f(x) passes through the origin. Lines drawn parallel to the coordinate axes through an arbitary point of the curve form a rectangle with two sides on the axes. The curve divides every such rectangle into two regions A and B, one of which has an area equal to n times the other. Find the functionf. 6. (a) Let u be a nonzero solution of the second-order equation y" + P(x)/ + Q(x)y = 0. Show that the substitution y = uv converts the equation y" + P(x),' + Q(x,y = R(x) into a first-order linear equation for v' . (b) Obtain a nonzero solution of the equation y' - 4y' + x2(y' - 4y) = 0 by inspection and use the method of part (a) to find a solution of yfl - 4y' + x2 (y' - 4y) = 2xe-x3/3 suchthaty=Oandy'=4whenx=O. Linear equations ofsecond order with constant coeflcients. In each of Exercises 7 through 10, find all solutions on ( - lm, + m). 7. yn - 4 y = o . 9. yn -2y'+sy =o. 8. y" + 4y = 0. 10. y" +2y'+y=o.11. Find all values of the constant k such that the differential equation y" + ky = 0 has a nontrivial solution y =f&) for which&(O) =fk(l) = 0. For each permissible value of k, determine the corresponding solution y =fk(x) . Consider both positive and negative values of k.12. If (a, b) is a given point in the plane and if m is a given real number, prove that the differential equation y" + k"y = 0 has exactly one solution whose graph passes through (a, b) and has slope m there. Discuss separately the case k = 0.13. In each case, find a linear differential equation of second order satisfied by ur and u2. (a> ul(x) = e" , u2(x) = e-". (b) ul(x) = e2x, u2(x) = xe2z. (c) ul(x) = eP'2 cos x, u2(x) = eh2 sin x. (d) z+(x) = sin (2x + l), u2(x) = sin (2x + 2). (e) ul(x) = cash x, u2w = sinh x .14. A particle undergoes simple harmonic motion. Initially its displacement is 1, its velocity is 1 and its acceleration is - 12. Compute its displacement and acceleration when the velocity is 4s.6.4 Linear differential equations of order n A linear differential equation of order n is one of the form(6.6) P,(x)y'n) + P1(x)y'"-1' + 9 * * + P,(x)y = R(x) *The functions P,, , P, , . . . , P, multiplying the various derivatives of the unknown functiony are called the coeficients of the equation. In our general discussion of the linear equationwe shall assume that all the coefficients are continuous on some interval J. The word"interval" will refer either to a bounded or to an unbounded interval.
146 Linear d@erential equations~_ In the differential equation (6.6) the leading coefficient P, plays a special role, since itdetermines the order of the equation. Points at which P,(x) = 0 are called singular pointsof the equation. The presence of singular points sometimes introduces complications thatrequire special investigation. To avoid these difficulties we assume that the function P,is never zero on J. Then we can divide both sides of Equation (6.6) by PO and rewrite thedifferential equation in a form with leading coefficient 1. Therefore, in our general dis-cussion we shall assume that the differential equation has the form(6.7) y@) + P,(x)y-l) + * * * + P,(x)y = R(x). The discussion of linear equations can be simplified by the use of operator notation. Let V(J) denote the linear space consisting of all real-valued functions continuous on an intervalJ. Let F"(J) denote the subspace consisting of all functions f whose first n derivativesj-',f", . I. ,ftn) exist and are continuous on J. Let P,, . . . , P, be n given functions in %?(J) and consider the operator L: V"(J) -+ V(J) given by L(f) = f (n) + PJ'"-1' + * * * + PJ.The operator L itself is sometimes written aswhere Dk denotes the kth derivative operator. In operator notation the differential equationin (6.7) is written simply as 1(6.8) L(y) = R.A solution of this equation is any function y in V'"(J) which satisfies (6.8) on the interval J. It is easy to verify that L(yl + yz) = L(yl) + L(y&, and that L(cy) = CL(~) for everyconstant c. Therefore L is a linear operator. This is why the equation L(y) = R is referredto as a linear equation. The operator L is called a linear d$erential operator of order n. With each linear equation L(y) = R we may associate the equation L(y) = 0,in which the right-hand side has been replaced by zero. This is called the homogeneousequation corresponding to L(y) = R. When R is not identically zero, the equation L(y) = Ris called a nonhomogeneous equation. We shall find that we can always solve the non-homogeneous equation whenever we can solve the corresponding homogeneous equation.Therefore, we begin our study with the homogeneous case. The set of solutions of the homogeneous equation is the null space N(L) of the operatorL. This is also called the solution space of the equation. The solution space is a subspaceof V:"(J). Although GF?(.~) .is infinite-dimensional, it turns out that the solution space N(L)is always finite-dimensional. In fact, we shall prove that(6.9) dim N(L) = n,
The dimension of the solution space of a homogeneous linear equation 147where n is the order of the operator L. Equation (6.9) is called the dimensionality theoremfor linear differential operators. The dimensionality theorem will be deduced as a conse-quence of an existence-uniqueness theorem which we discuss next.6.5 The existence-uniqueness theorem THEOREM 6.3. EXISTENCE-UNIQUENESS THEOREM FOR LINEAR EQUATIONS OF ORDER n. LetPl,P,, * * . , P, be continuous functions on an open interval J, alzd let L be the linear d@er-ential operator L=Dn+PIDn-l+...+P,. IfxOEJandifk,,k,,..., knpl are n given real numbers, then there exists one and only onefunction y = f(x) which satisfies the homogeneous dlrerential equation L(y) = 0 on J and which also satisfies the initial conditions f(x,) = k, ,f'(x,) = kl, . . . ,f +l)(x,,) = k,-, . Note: The vector in n-space given by (f(x,),f'(x,), . . . ,f(n-l)(~O)) is called the initial-value vector off at x0, Theorem 6.3 tells us that if we choose a point x0 in J and choose a vector in n-space, then the homogeneous equation L(y) = 0 has exactly one solution y = f (x) on J with this vector as initial-value vector at x0. For example, when n = 2 there is exactly one solution with prescribed value f(xo) and prescribed derivative f'(xo) at a prescribed point x0. The proof of the existence-uniqueness theorem will be obtained as a corollary of moregeneral existence-uniqueness theorems discussed in Chapter 7. An alternate proof for thecase of equations with constant coefficients is given in Section 7.9.6.6 The dimension of the solution space of a homogeneous linear equation THEOREM 6.4. DIMENSIONALITY THEOREM. Let L: F(J) + 9?(J) be a linear d@erentialoperator of order n given by(6.10) L = D" +PIDn-' + *.* +P,.Then the solution space of the equation L(y) = 0 has dimension n. Proof. Let V, denote the n-dimensional linear space of n-tuples of scalars. Let T be thelinear transformation that maps each function f in the solution space N(L) onto the initial-value vector off at x0, T(f) = (f(x,,>>f 'Cd, . . . J-'"-l'(~,>) 3where x,, is a fixed point in J. The uniqueness theorem tells us that T(f) = 0 implies f = 0.Therefore, by Theorem 2.10, T is one-to-one on N(L). Hence T-l is also one-to-one andmaps V, onto N(L)!. and Theorem 2.11 shows that dim N(L) = dim V, = n .
148 Linear diferential equations Now that we know that the solution space has dimension n, any set of n independentsolutions will serve as a ibasis. Therefore, as a corollary of the dimensionality theorem wehave : THEOREM 6.5. Let L: 'g"(J) -+ V(J) be a linear dtyerential operator of order n. If ul, . . . ,u, are n independent solutions of the homogeneous differential equation L(y) = 0 on J, thenevery solution y = f (x) on J can be expressed in the form(6.11)where cl, . . . , c, are constants. Note: Since all solutions of the differential equation L(y) = 0 are contained in formula (6.1 I), the linear combination on the right, with arbitrary constants cl, . . . , c,, is sometimes called the general solution of the differential equation. The dimensionality theorem tells us that the solution space of a homogeneous lineardifferential equation of order n always has a basis of n solutions, but it does not tell us howto determine such a basis. In fact, no simple method is known for determining a basis ofsolutions for every linear equation. However, special methods have been devised forspecial equations. Among these are differential equations with constant coefficients towhich we turn now.6.7 The algebra of constant-coefficient operators A constant-coefficient operator A is a linear operator of the form(6.12) .4 = a,,Dn + aIDn-l + * * * + a,-,D + a,,where D is the derivative operator and a,, a,, . . . , a, are real constants. If a, # 0 theoperator is said to have order n. The operator A can be applied to any function y with nderivatives on some interval, the result being a function A(y) given by A(y) = a,,y(") + aIy(+l) + * * * + a,-,y' + any.In this section, we restrict our attention to functions having derivatives of every order on(- co, + co). The set of all such functions will be denoted by 59" and will be referred to asthe class of injnitely di#erentiable functions. If y E ?Zrn then A(y) is also in Ym. The usual algebraic operations on linear transformations (addition, ,multiplication byscalars, and composition or multiplication) can be applied, in particular, to constant-coefficient operators. Let A and B be two constant-coefficient operators (not necessarilyof the same order). Since the sum A + B and all scalar ,multiples IA are also constant-coefficient operators, the set of all constant-coefficient operators is a linear space. Theproduct of A and B (in either order) is also a constant-coefficient operator. Therefore,sums, products, and scalar multiples of constant-coefficient operators satisfy the usualcommutative, associative, and distributive laws satisfied by all linear transformations.Also, since we have DrDs = DSDT for all positive integers r and s, any two constant- coefficient operators commute; AB = BA .
The algebra of constant-coejicient operators 149 With each constant-coefficient operator A we associate a polynomial pA4 called thecharacteristic polynomial of A. If A is given by (6.12),p, is that polynomial which has thesame coefficients as A. That is, for every real r we have PA(r) = a/ + aIF + * * * + a,.Conversely, given any real polynomial p, there is a corresponding operator A whosecoefficients are the same as those ofp. The next theorem shows that this associationbetween operators and polynomials is a one-to-one correspondence. Moreover, thiscorrespondence associates with sums, products, and scalar multiples of operators therespective sums, products, and scalar multiples of their characteristic polynomials. THEOREM 6.6. Let A and B denote constant-coeficient operators with characteristicpolynomials pA and pB , respectively, and let 1 be a real number. Then ~'e have: (C) PAB~~~:BY (4 PAA * Proof. We consider part (a) first. AssumepA = pB . We wish to prove that A(J) = B(y)for every y in v". Since pA = pB, both polynomials have the same degree and the samecoefficients. Therefore A and B have the same order and the same coefficients, so A(y) =B(y) for every y in V". Next we prove that A = B impliesp, = pB . The relation A = B means that A(y) = B(y)for every y in %?a. Take y = er", where r is a constant. Since y(l;) = rkerr for every k 2 0,we have 4.~9 = pA(de'" and B(y) = pB(r)e".The equation A(y) = B(j) implies pA4(r) = pB(r). S ince r is arbitrary we must havepA = pB . This completes the proof of part (a). Parts (b), (c), and (d) follow at once from the definition of the characteristic polynomial. From Theorem 6.6 it follows that every algebraic relation involving sums, products, andscalar multiples of polynomials pA4 and pB also holds for the operators A and B. Inparticular, if the characteristic polynomial pAI can be factored as a product of two or morepolynomials, each factor must be the characteristic polynomial of some constant-coefficientoperator, so, by Theorem 6.6, there is a corresponding factorization of the operator A.For example, ifp,(r) = pB(r)pc(r), then A = BC. Ifpe4(r) can be factored as a product ofn linear factors, say(6.13) h(r) = a& - rl)@ - r2) - * * (r - IT,),the corresponding factorization of A takes the form A = a,(D - r,)(L) - r2) * * * (D - rJ.
Determination of a basis of solutions for linear equations 151Therefore the null space of L contains the null space of the last factor A,. But sinceconstant-coefficient operators commute, we can rearrange the factors so that any one ofthem is the last factor. This proves (6.14). If L(u) = 0, the operator L is said to annihilate U. Theorem 6.7 tells us that if a factor Aiof L annihilates u:, then L also annihilates U. We illustrate how the theorem can be used to solve homogeneous differential equationswith constant coefficients. We have chosen examples to illustrate different features,depending on the nature of the roots of the characteristic equation. CASE I. Real distinct roots. EXAMPLE 1. Find a basis of solutions for the differential equation(6.15) (D3 - 70+6)y=O. Solution. This lhas the form L(y) = 0 with L =D3-7D+6=(D-l)(D-2)(0+3).The null space of 11 - 1 contains ur(x) = t?, that of D - 2 contains uZ(x) = ezZ, and thatof D + 3 contains u3(x) = e-3x. In Chapter 1 (p. 10) we proved that u1 , u2, us are inde-pendent. Since three independent solutions of a third order equation form a basis for thesolution space, the general solution of (6.15) is given by Y = cleZ + c2e2" + c3ew3'. The method used to solve Example 1 enables us to find a basis for the solution space ofany constant-coefficient operator that can be factored into distinct linear factors. THEOREM 6.8. Let L be a constant coeflcient operator whose characteristic equationpL(r) = 0 has n distinct real roots rI , r2, . . . , r, . Then the general solution of the difSerentia1equation L(y) = 0 on the interval (- co, + KI) is given by the formula(6.16) y = j$ ckerk" . k=l Proof. We have the factorization L = a,(D - rI)(D - r2). * * (D - r,).Since the null space of (D - rJ contains uk(x) = erkr, the null space of L contains the nfunctions(6.17) U1(x) = f?, uz(X) = er@, . . . , u,(x) = eTnr,
152 Linear d@erential equationsIn Chapter 1 (p. 10) we proved that these functions are independent. Therefore they forma basis for the solution space of the equation L(y) = 0, so the general solution is given by(6.16). CASE II. Real roots, some of which are repeated. If all the roots are real but not distinct, the functions in (6.17) are not independent andtherefore do not form a basis for the solution space. If a root r occurs with multiplicity m,then (D - r)"' is a factor of L. The next theorem shows how to obtain m independentsolutions in the null space of this factor. THEOREM 6.9. The m jimctions ul(x) = e'", u2(x) = xerr, . . . , u,(x) = xm-%?*are m independent elements annihilated by the operator (D - r)" . Proof. The independence of these functions follows from the independence of thepolynomials 1, x, x2, . . . , x+-l. To prove that ur, u2, . . . , u, are annihilated by (D - r)"we use induction on m. If m = 1 there is only one function, ul(x) = eF2, which is clearly annihilated by (D - r) .Suppose, then, that the theorem is true for m - 1. This means that the functions ul, . . , ,u,-~ are annihilated by (D - r)m-l. Since (D - r)" = (D - r)(D - r)+'the functions u1 , . . . , u,,-~ are also annihilated by (D - r)" . To complete the proof wemust show that (D - r)m annihilates u, . Therefore we consider (D - r)%, = (D - r)m-l(D - r)(xm-1e'3c).We have (D - r)(x"-le'") = D(x"-lp) - rxm-leTz = (m - 1)xm--2p + xm-lr@x - rx~-lp = (m - 1)xm-2erz = (m - l)u,-,(x).When we apply (D - r:Y'-' to both members of this last equation we get 0 on the rightsince (D - r)m-l annihilates u,-r . Hence (D - r)%, = 0 so u, is annihilated by(D - r)". This completes the proof. EXAMPLE 2. Find the general solution of the differential equation L(y) = 0, whereL=D3-D2-8D+12. Solution. The operator L has the factorization L = (D - 2)2(D + 3).
Determination of a basis of solutions for linear equations 153By Theorem 6.9, the two functions q(x) = e2* ) z&(x) = xe2=are in the null space of (D - 2)2. The function us(x) = e+ is in the null space of (D + 3).Since ul, u2, ua are independent (see Exercise 17 of Section 6.9) they form a basis for thenull space of L, so the general solution of the differential equation is y = cle2s + c2xe2x + c3ee3S. Theorem 6.9 tells us how to find a basis of solutions for any nth order linear equationwith constant coefficients whose characteristic equation has only real roots, some of whichare repeated. If the distinct roots are rl, r2, . . . , r, and if they occur with respectivemultiplicities m, , m2, . . . , mk, that part of the basis corresponding to rs is given by the m,functions u,,,(x) = x*-leQr, where q= I,2 ,..., mM,.Asptakesthevalues1,2,...,kwegetm,+*** + mk functions altogether. In Exercise17 of Section 6.9 we outline a proof showing that all these functions are independent. Sincethe sum of the multiplicities m, + * * * + m, is equal to n, the order of the equation, thefunctions u form a basis for the solution space of the equation. B,Q EXAMPLE 3. Solve the equation (D6 + 2D5 - 2D3 - D2)y = 0. Solution. We have D6 + 2D5 - 2D3 - D2 = D2(D - l)(D + I)". The part of thebasis corresponding to the factor D2 is ul(x) = 1, u2(x) = x ; the part corresponding tothe factor (D - 1) is u3(x) = e" ; and the part corresponding to the factor (D + 1)" isu4(x) = e-", us(x) = xe-" , Us = x2ePr. The six functions ul, . . . , us are independentso the general solution of the equation is y = cl + c2x + c3e5 + (c4 + c,x + c6x2)emz. CASE III. Complex roots. If complex exponentials are used, there is no need to distinguish between real and complexroots of the characteristic equation of the differential equation L(y) = 0. If real-valuedsolutions are desired, we factor the operator L into linear and quadratic factors with realcoefficients. Each pair of conjugate complex roots u + i,!?, u - i/3 corresponds to aquadratic factor,(6.18) D2-2uD+cc2+/12.The null space of this second-order operator contains the two independent functionsU(X) = eaz cos px and v(x) = ear sin px. If the pair of roots dc f i/3 occurs with multi-plicity m, the quadratic factor occurs to the mth power. The null space of the operator [D2 - 2uD + u2 + p21rn
156 Linear d@erential equations19. Refer to the notation of E.xercise 18. If u and u have n derivatives, prove that k=O [Hint: Use Exercise 18 along with Leibniz's formula for the kth derivative of a product: (m)(k) =r~o(:,u(k-w~) .]20. (a) I&p(t) = q(t)?(t), where q and r are polynomials and m is a positive integer. Prove that p'(t) = q(t)"-is(t), where s is a polynomial. (b) Let 1, be a constant-coefficient operator which annihilates u, where u is a given function of x . LetM=Lm,the mth power of L, where m > 1. Prove that each of the derivatives M', M", . . . ) M("-l) also annihilates u. (c) Use part (b) and Exercise 19 to prove that M annihilates each of the functions u, xu, . . . , xm-Ill. (d) Use part (c) to show that the operator (D2 - 2ctD + a2 + p2)" annihilates each of the functions xQe"z sin px and x*eaz cos px for q = 1, 2, . . . , m - 1 .21. Let L be a constant-coefficient operator of order n with characteristic polynomial p(r). If c( is constant and if u has n derivatives, prove that L(e"%(x)) = eaz n p? U(k)(X). c . k=O6.10 The relation between the homogeneous and nonhomogeneous equations We return now to the general linear differential equation of order n with coefficients thatare not necessarily constant. The next theorem describes the relation between solutions ofa homogeneous equation L(;v) = 0 and those of a nonhomogeneous equation L(y) = R(x). THEOREM 6.10. Let L 9*(J) * V(J) be a linear difSerentia1 operator of order n. Let ul, . . . ,u, be n independent solutions of the homogeneous equation L(y) = 0, and let yl be a particularsolution of the nonhomogeneous equation L(y) = R, where R E V(J). Then every solutiony = f (x) of the nonhomogeneous equation has the form(6.19) .I-@> = Yl(X> +~ykwwhere cl, . . . , c, are constants. Proof. By linearity we have L(f - yr) = L(f) - L(yr) = R - R = 0. Thereforef - y1 is in the solution space of the homogeneous equation L(y) = 0, so f - y1 is a linearcombination of ul, . . . , u, , sayf - y, = clul + . . - + c,u, . This proves (6.19). Since all solutions of L(y) = R are found in (6.19), the sum on the right of (6.19) (witharbitrary constants cl, c2, . . . , c,) is called the general solution of the nonhomogeneous
Determination of a particular solution of the nonhomogeneous equation 157equation. Theorem 6.10 states that the general solution of the nonhomogeneous equationis obtained by adding toy, the general solution of the homogeneous equation. Note: Theorem 6.10 has a simple geometric analogy which helps give an insight into its meaning. To determine all points on a plane we find a particular point on the plane and add to it all points on the parallel plane through the origin. To find all solutions of L(y) = R we find a particular solution and add to it all solutions of the homogeneous equation L(y) = 0. The set of solutions of the nonhomogeneous equation is analogous to a plane through a particular point. The solution space of the homogeneous equation is analogous to a parallel plane through the origin. To use Theorem 6.10 in practice we must solve two problems: (1) Find the generalsolution of the homogeneous equation L(y) = 0, and (2) find a particular solution of thenonhomogeneous equation L(y) = R. In the next section we show that we can alwayssolve problem (2) if we can solve problem (1).6.11 Determination of a particular solution of the nonhomogeneous equation. The method of variation of parameters We turn now to the problem of determining one particular solution yr of the nonhomo-geneous equation L(y) = R . We shall describe a method known as variation ofparameterswhich tells us how to determine y1 if we know n independent solutions ul, . . . , u, of thehomogeneous equation L(y) = 0. The method provides a particular solution of the form(6.20) y1= VI%+ --- + v,u,,wherev,, . . . , v, are functions that can be calculated in terms of u1 , . . . , u, and the right-hand member R. The method leads to a system of n linear algebraic equations satisfied bythe derivatives vi, . . . , v:. This system can always be solved because it has a nonsingularcoefficient matrix. Integration of the derivatives then gives the required functions vi, . . . ,v, . The method was first used by Johann Bernoulli to solve linear equations of first order,and then by Lagrange in 1774 to solve linear equations of second order. For the nth order case the details can be simplified by using vector and matrix notation.The right-hand member of (6.20) can be written as an inner product,(6.21) Yl = (0, 4,where v and u are n-dimensional vector functions given by v = (Vl, - * * 9 %I, u = (Ul,. . . ) UJ.We try to choose v so that the inner product defining y1 will satisfy the nonhomogeneousequation L(y) = R, given that L(u) = 0, where L(u) = (L(u,), . . . , L(u,)). We begin by calculating the first derivative of yr . We find(6.22) y; = (0, u') + (u', u).
158 Linear d@erential equationsWe have n functions vl, . . . , v, to determine, so we should be able to put n conditions onthem. If we impose the condition that the second term on the right of (6.22) should vanish,the formula for yi simplifies to y; =: (v, U')) provided that (v', u) = 0.Differentiating the relation for y; we find yf = (v, 24") + (v', u') .If we can choose v so that (Iv', u') = 0 then the formula for yy also simplifies and we get y; = (v, u")) provided that also (v', u') = 0.If we continue in this manner for the first n - 1 derivatives of y1 we find ,:,-1) = (v, &L-l))) provided that also (v', u(+')) = 0.So far we have put n - 1 conditions on v. Differentiating once more we get y:"' = (v, ZP) + (v', ZP-1)).This time we impose the condition (v', u @-l)) = R(x), and the last equation becomes y:"' = (v, P) + R(X)) provided that also (v', u(~-')) = R(x).Suppose, for the moment, that we can satisfy the n conditions imposed on v. Let L =D" + P,(x)D"-l + * . . + P,(x). When we apply L to y1 we find I&) = y:"' + P,(x)Jpl' + . . . + P,(X>Yl = {(u, U(n:' ) + R(x)} + Pl(X)(U, LP-l)) + . . . + P,(x)(u, u) = (v, L(u)) + R(x) = (v, 0) + R(x) = R(x).Thus L(yJ = R(x), so y1 is a solution of the nonhomogeous equation. The method will succeed if we can satisfy the n conditions we have imposed on v. Theseconditions state that (v', u("') = 0 for k = 0, 1, . . . , n - 2, and that (v', zJn-l)) = R(x).We can write these n equations as a single matrix equation, - - 0(6.23) W(x)v'(x) = R(x) i II 1
Determination of a particular solution of the nonhomogeneous equation 159where v'(x) is regarded as an n x 1 column matrix, and where W is the n x n matrixfunction whose rows consist of the components of u and its successive derivatives: % l.lz ..* un 4 4 ... 11; w= . . (n-1) (n-1) . . . U(n-l) -Ul uz .I nThe matrix W is called the Wronskian matrix of ul, . . . , u,, after J. M. H. Wronski u'(x)=R(x)W(xI)-l0 .(1778-1853). In the next section we shall prove that the Wronskian matrix is nonsingular. Thereforewe can multiply both sides of (6.23) by W(x)-l to obtain 0 1Choose two points c and x in the interval J under consideration and integrate this vectorequation over the interval from c to x to obtain u(x) = u(c) +whereThe formula y1 = (u, v) for the particular solution now becomes y1 = (4 v) = (%0(C) + 2) = (t&u(c)) + (u,z).The first term (u, v(c)) satisfies the homogeneous equation since it is a linear combination of241,. . . ) u,. Therefore we can omit this term and use the second term (u, z) as a particularsolution of the nonhomogeneous equation. In other words, a particular solution of
160 Linear dtrerential equationsL(y) = R is given by the inner product 0 (u(x), z(x)> = (u(x),lZR(t)W(t)-' ; dt) . 1 ;INote that it is not necessary that the function R be continuous on the interval J. All that isrequired is that R be integrable on [c, x]. We can summarize the results of this section by the following theorem. THEOREM 6.11. Let Ml,... , u, be n independent solutions of the homogeneous nth orderlinear dtxerential equation L,(y) = 0 on an interval J. Then a particular solution y1 of thenonhomogeneous equation L(y) = R is given by the formulawhere vl, . . . , v, are the entries in the n x 1 column matrix v determined-by the equation(6.24) sR(x)W(x)-' I;j dx.In this formula, W is the Wronskian matrix of ul, . . . , u,, and c is any point in J. Note: The definite integral in (6.24) can be replaced by any indefinite integral 0 1 EXAMPLE 1. Find the general solution of the differential equation ))" - y = 2 1 + e"on the interval (- co, + co). Solution. The homogeneous equation, (D2 - 1)y = 0 has the two independent solutionsul(x) = eZ, uZ(x) = e-" . The Wronskian matrix of u1 and u2 is ez e-z W(x) = [ e" -e-" * 1
The annihilator method for determining a particular solution 1636.13 Special methods for determining a particular solution of the nonhomogeneous equation. Reduction to a system of first-order linear equations Although variation of parameters provides a general method for determining a particularsolution of ,5,(y) = R, special methods are available that are often easier to apply when theequation has certain special forms. For example, if the equation has constant coefficientswe can reduce the problem to that of solving a succession of linear equations of first order.The general method is best illustrated with a simple example. EXAMPLE 1. Find a particular solution of the equation(6.28) (D - l)(D - 2)~ = xe"+"' . Solution. Let u = (D - 2)~. Then the equation becomes (D - 1)~ = xez+"' .This is a first-order linear equation in u which can be solved using Theorem 6.1. A particularsolution is u = $es+x2Substituting this in the equation u = (D - 2)~ we obtaina first-order linear equation for y. Solving this by Theorem 6.1 we find that a particularsolution (withy,(O) = 0) is given by yl(x) = frezr ," et'-' dt . IAlthough the integral cannot be evaluated in terms of elementary functions we considerthe equation as having been solved, since the solution is expressed in terms of integrals offamiliar functions. The general solution of (6.28) is y = cle2 + c2ezx + +ezx * et2-tdt. i06.14 The annihilator method for determining a particular solution of the nonhomogeneous equation We describe next a method which can be used if the equation L(y) = R has constantcoefficients and if the right-hand member R is itself annihilated by a constant-coefficientoperator, say A(R) = 0. In principle, the method is very simple. We apply the operatorA to both members of the differential equation L(y) = R and obtain a new equationAL(y) = 0 which must be satisfied by all solutions of the original equation. Since AL isanother constant-coefficient operator we can determine its null space by calculating theroots of the characteristic equation of AL. Then the problem remains of choosing from
Miscellaneous exercises on linear diflerenfial equations 16713. Given two constant-coefficient operators A and B whose characteristic polynomials have no zeros in common. Let C = AB. (a) Prove that every solution of the differential equation C(y) = 0 has the form y = y1 + ys, where A(yJ = 0 and B(yJ = 0. (b) Prove that the functions yr and ye in part (a) are uniquely determined. That is, for a given y satisfying C(y) = 0 there is only one pair y1 , yz with the properties in part (a).14. If L(y) = y" + uy' + by, where u and b are constants, let f be that particular solution of L(y) = 0 satisfying the conditions f(0) = 0 and f'(0) = 1 . Show that a particular solution of L(y) = R is given by the formula for any choice of c. In particular, if the roots of the characteristic equation are equal, say rl = r2 = m , show that the formula for y,(x) becomes yl(x) = ema - t)epmtR(t) dt.15. Let Q be the operator "multiplication by x." That is, Q(y)(x) = x . y(x) for each y in class %" and each real x. Let Z denote the identity operator, defined by Z(y) = y for each y in %?". (a) Prove that DQ - QZ) = I. (b) Show'that D2Q - QDz is a constant-coefficient operator of first order, and determine this operator explicitly as a linear polynomial in D. (c) Show that D3Q - QD3 is a constant-coefficient operator of second order, and determine this operator explicitly as a quadratic polynomial in D. (d) Guess the generalization suggested for the operator D"Q - QDn, and prove your result by induction.In each of Exercises 16 through 20, find the general solution of the differential equation in thegiven interval.16. y" -y = l/x, (0, +a).17. y" + 4y = set 2x, (- $ , a1. 7-r 7718. y" -y = sec3x - secx, ( -Tj,j 1 *19. y" - 2y' + y = ee"(ez - 1)2, (-co, +a).20. y" - 7y" + 14y' - sy = log x, (0, +a).6.16 Miscellaneous exercises on linear differential equations 1. An integral curvey = U(X) of the differential equation yfl - 3~1 - 4y = 0 intersects an integral curve y = v(x) of the differential equation y"+ 4y' - 5y = 0 at the origin. Determine the functions u and v if the two curves have equal slopes at the origin and if [a414 5 ;y&j- =6. 2. An integral curve y = u(x) of the differential equation y" - 4y' + 29y = 0 intersects an integral curve y = v(x) of the differential equation y" + 4y' + 13y = 0 at the origin. The two curves have equal slopes at the origin. Determine u and v if u'(rr/2) = 1 .
168 Linear d@erential equations 3. Given that the differential equation y" + 4xy' + Q(x)y = 0 has two solutions of the form '0; ; u(x) and y2 = 4x>, where u(0) = 1 . Determine both u(x) and Q(x) explicitly in terms 4. Let L(y) = y' + Ply' + P,y . To solve the nonhomogeneous equation L(y) = R by variation of parameters, we need to know two linearly independent solutions of the homogeneous equation. This exercise shows that if one solution aI of L(y) = 0 is known, and if aI is never = 4(x)sc '[UQ(l(t)1t)2 dt 9 zero on an interval J, a second solution u2 of the homogeneous equation is given by the formula u2(4 where Q(x) = ~~r'l(~)~~, and c is any point in J. These two solutions are independent on J. (a) Prove that the function u2 does, indeed, satisfy L(y) = 0. (b) Prove that a1 and u2 are independent on J. 5. Find the general solution of the equation xy " - 2(x + 1)~' + (X + 2)y = x3e2~ for x > 0, given that the homogeneous equation has a solution of the form y = emr . 6. Obtain one nonzero solution by inspection and then find the general solution of the differential equation (y" - 4y') + x2(y' - 4y) = 0. 7. Find the general solution of the differential equation 4x2y" +4xy'-y =o, given that there is a particular solution of the form y = x" for x > 0. 8. Find a solution of the homogeneous equation by trial, and then find the general solution of the equation x(1 - xI)y" - (1 - 2x)y' + (x2 - 3x + 1)y = (1 - x)3. 9. Find the general solution of the equation (2x - 3x3)y" + 4y' + 6xy = 0, given that it has a solution that is a polynomial in x.10. Find the general solution of the equation x2( 1 - x)y" + 2x(2 - x)y' + 2(1 + x)y = x2 ) given that the homogeneous equation has a solution of the form y = XC.11. Let g(x) = JT et/t dt if x > 0. (Do not attempt to evaluate this integral.) Find all values of the constant a such that the function f defined by f(x) = ieag(z)
Linear equations of second order with analytic coefficients 169 satisfies the linear differential equation x2yn + (3x - x2)/ + (1 - x - e2,c)y = 0. Use this information to determine the general solution of the Iequation on the interval (0, + to).6.17 Linear equations of second order with analytic coeflicients A functionfis said to be analytic on an interval (x0 - r, x0 + r) iffhas a power-seriesexpansion in this interval,convergent for Ix - x0] < r. If the coefficients of a homogeneous linear differentialequation y@) + P1(x)y(-) + - * * + P,(x)y = 0are analytic in an interval (x,, - r, x,, + r), then it can be shown that there exist n inde-pendent solutions ul, . . . , u,, each of which is analytic o.n the same interval. We shallprove this theorem for equations of second order and then discuss an important examplethat occurs in many applications. THEOREM 6.13. Let P, and P, be analytic on an open interval (x, - r, x0 + r), say P,($ =n~oUx - XOY 3 P2(x) = i c;,(x - XJ . 12=0Then the deferential equation(6.31) y" + P,(x)y' + P2<4y = 0has two independent solutions u1 and u2 which are analytic on the same interval. Proof. We try to find a power-series solution of the form(6.32) y = 2 a& - xo)lz, ?I=0convergent in the given interval. To do this, we substitute the given series for P, and P, inthe differential equation and then determine relations which the coefficients a,, must satisfyso that the function y given by (6.32) will satisfy the equation. The derivatives y' and y" can be obtained by differentiating the power series for y termby term (see Theorem 11.9 in Volume I). This gives us y' =$;a& - xoY--l =zJn + l)a,+lb - -WY y" =z;(n - l)a,(x - x0)+' =gJn + 2)(n -I- l)a,+,(x - xoY.
170 Linear d@erential equationsThe products P,(x)y' and Pz(x)v are given by the power series?andWhen these series are substituted in the differential equation (6.31) we find n + 2)(n + l)a,+, + i Kk + 1) Uk+lbn-k + a,&-,] (x - &J)~ = 0. k=O ITherefore the differential equation will be satisfied if we choose the coefficients a, so thatthey satisfy the recursion formula(6.33) (n + 2)(n + lb,,, = -$Jck + l)uk+lbn-k -k akcn-klfor n=0,1,2 ,.... This formula expresses an+2 in terms of the earlier coefficientsao,~l,...,~,+l and the coefficients of the given functions P, and P2. We choose arbitraryvalues of the first two coefficients a, and a, and use the recursion formula to define theremaining coefficients a2, u3, . . . , in terms of a, and a,. This guarantees that the powerseries in (6.32) will satisfy the differential equation (6.31). The next step in the proof is toshow that the series so defined actually converges for every x in the interval (x0 - r, x0 + r).This is done by dominating the series in (6.32) by another power series known to converge.Finally, we show that we can choose a, and a, to obtain two independent solutions. We prove now that the series (6.32) whose coefficients are defined by (6.33) converges inthe required interval. Choose a fixed point x1 # x0 in the interval (x0 - r, x0 + r) and let t = Ix1 - x01.Since the series for P, and P, converge absolutely for x = x1 the terms of these series arebounded, say @ki tk 5 Ml and icki tk 5 M,~for some MI > 0, Mz > 0. Let A4 be the larger of M, and tM, . Then we haveThe recursion formula implies the inequality (n + 2)(n + 1) la n+sl $& + 1) bk+ll Fk + bkl 5) k=O M =- n+l n (k + 1) bk+ll tk+'+ &,I/ t"+' + boi - b,+,l tn+l t lx k=O 1 k=O I j$ -&+ 2) Ia k+ll tk+' + luol = -$ T@ + 1) l&l tk. k=O I k=Ot Those readers not familiar with multiplication of power series may consult Exercise 7 of Section 6.21.
The Legendre polynomials 175where [n/2] denotes the greatest integer 5 n/2. We will show presently that this is theLegendrepolynomial of degree n introduced in Chapter 1. When n is even, it is a constantmultiple of the polynomial ul(x) in Equation (6.40); when n is odd, it is a constant multipleof the polynomial uz(x) in (6.41).? The first seven Legendre polynomials are given by theformulas Po(x> = 1 7 PI(X) = x, P2(x) = &(3x2 - I), P3(x) = $(5x3 - 3x), PJ(X) = $(35x4 - 30x2 + 3), P5(x) = &(63x5 - 70x3 + 15x), , P6(x) = ,+(231x6 - 315x4 + 105x2 - 5).Figure 6.1 shows the graphs of the first five of these functions over the interval [-I , I]. Y P2 P, P, --I FIGURE 6.1 Graphs of Legendre polynomials over the interval [ - 1 , 11. Now we can show that, except for scalar factors, the Legendre polynomials are thoseobtained by applying the Gram-Schmidt orthogonalization process to the sequence ofpolynomials 1, x, x2, . . . , with the inner productt When n is even, say n = 2m, we may replace the index of summation k in Equation (6.40) by a new indexr, where Y = m - k ; we find that the sum in (6.40) is a constant multiple of P,(x). Similarly, when n isodd, a change of index transforms the sum in (6.41) to a constant multiple of P,(x).
176 Linear dlflerential equations First we note that if m # n the polynomials P, and P, are orthogonal because they areeigenfunctions of a symmetric operator belonging to distinct eigenvalues. Also, since P,has degree n and P,, = 1, the polynomials P,(x), PI(x), . . . , P,(x) span the same subspaceas 1,x ,..., x". In Section 1.14, Example 2, we constructed another orthogonal set ofpolynomials y, , y, , y,, . . . , such that y,(x), ul(x), . . . , yJx> spans the same subspace as1,x,..., x" for each n. The orthogonalization theorem (Theorem 1.13) tells us that,except for scalar factors, there is only one set of orthogonal functions with this property.Hence we must have P,(x) = w,(x)for some scalars c, . The coefficient of xn in y,(x) is 1, so c, is the coefficient of xn in P,(x).From (6.42) we see that c _ On)! n 2"(n !)2 .6.20 Rodrigues' formula for the Legendre polynomials In the sum (6.42) defining P,(x) we note that d" (2n-22r)! xn _ 2r= - X2n-2P and 1 =- n 1 (n - 2r)! dx" r! (n - r)! n! 0 r 'where (;) is the binomial coefficient, and we write the sum in the form P,(x) = & j.g (4)'(;)x2"2'. r=oWhen [n/2] < r 5 n, the term x2n-2r has degree less than n, so its nth derivative is zero.Therefore we do not alter the sum if we allow r to run from 0 to n. This gives us p,(x) = 1 dn -+ 2"n! dx" /+(- 1)' ", X2n-2r- 7=0 0Now we recognize the sum on the right as the binomial expansion of (x2 - 1)". Thereforewe haveThis is known as Rodrigues'formula, in honor of Olinde Rodrigues (17941851), a Frencheconomist and reformer. Using Rodrigues' formula and the differential equation, we can derive a number ofimportant properties of the Legendre polynomials. Some of these properties are listedbelow. Their proofs are outlined in the next set of exercises. For each n 2 0 we have P,(l) = 1.
Exercises 177Moreover, P,(x) is the only polynomial which satisfies the Legendre equation (1 - x2)yC - 24 + n(n + lly = 0and has the value 1 when x = 1. For each n 2 0 we have P,(-x) = (-l)"P,(x).This shows that P, is an even function when n is even, and an odd function when n is odd. We have already mentioned the orthogonality relation, s-1, P,(x)P,(x) dx = 0 i f m#n.When m = n we have the norm relation lIPnIl = s-1 l [P,(x)]' dx = & . Every polynomial of degree n can be expressed as a linear combination of the Legendrepolynomials PO, P, , . . . , P, . In fact, iff is a polynomial of degree n we havewhere From the orthogonality relation it follows that s1, g(x)P,(x) dx = 0for every polynomial g of degree less than n. This property can be used to prove that theLegendre polynomial P, has n distinct real zeros and that they all lie in the open interval(-1 , 1).6.21 Exercises 1. The Legendre equation (6.35) with cc = 0 has the polynomial solution ul(x) = 1 and a solution u2, not a polynomial, given by the series in Equation (6.41). (a) Show that the sum of the series for u2 is given by 1 +x G4 = ; log rx for lx] X 1. (b) Verify directly that the function u2 in part (a) is a solution of the Legendre equation when a =o.
178 Linear d@erential equations2. Show that the function f defined by the equation f(X) = 1 - ; log s for 1x1 < 1 satisfies the Legendre equation (6.35) with G( = 1 . Express this function as a linear combination of the solutions ur and u2 given in Equations (6.38) and (6.39).3. The Legendre equation (6.35) can be written in the form [(x2 - l)y']' - cc(c( + 1)y = 0. (a) If a, b, c are constants with a > b and 4c + 1 > 0, show that a differential equation of the type [(x - 4(x - b)y']' - cy = 0 can be transformed to a Legendre equation by a change of variable of the form x = At + B, with A > 0. Determine A and B in terms of a and b. (b) Use the method suggested in part (a) to transform the equation (x2 - x)f + (2x - I)/ - 2r = 0 to a Legendre equation. 4. Find two independent power-series solutions of the Hermite equation y" - 2xy' f 2ary = 0 on an interval of the form (-r , r). Show that one of these solutions is a polynomial when cc is a nonnegative integer. 5. Find a power-series solution of the differential equation xy" + (3 + xyy + 3xzy = 0 valid for all x. Find a second solution of the form .y = x? 2 a,x" valid for all x # 0. 6. Find a power-series solution of the differential equation x2/' + x2/ - (ax + 2)y = 0 valid on an interval of the form (-r , r) . 7. Given two functions A and B analytic on an interval (x0 - r , x0 + r) , say A(x) = 2 a,(x - x0)%, B(x) = 2 b,(x - x0)". 71=0 n-0 It can be shown that the product C(x) = A(x)B(x) is also analytic on (x0 - r , x0 + r). This exercise shows that C has the power-series expansion C(x) = 2 c,(x - x0)", where c, =$oaA-a. T&=0
180 Linear d@erential equations11. (a) Show that P,(x) = &$ X" + Q,(x), where Q,(x) is a polynomial of degree less than n. (b) Express the polynomialf(x) = x4 as a linear combination of P,, , P, , Pz , P, , and P4. (c) Show that every polynomialf of degree II can be expressed as a linear combination of the Legendre polynomials P,,, P,, . . . , P,.12. (a) Iffis a polynomial of degree II, write [This is possible because of Exercise 11 (c).] For a fixed m, 0 5 m 5 n , multiply both sides of this equation by P,(x) and integrate from - 1 to 1. Use Exercises 9(b) and 10(b) to deduce the relation 2m+l l c, = - pxn(x) dx. 2 s13. Use Exercises 9 and 11 to show that ~~,g(x)P,(x) dx = 0 for every polynomial g of degree less than IZ.14. (a) Use Rolle's theorem to show that P, cannot have any multiple zeros in the open interval (-1 , 1). In other words, any zeros of P, which lie in (-1 , 1) must be simple zeros. (b) Assume P, has m zeros in the interval (- 1 , 1). If m = 0, let QO(x) = 1. If m 2 1, let Q,W = (x - x1)(x - x2) . . . (x - xm), wherex,,x,, . . . , x,arethemzerosofP,in(-1, 1). Showthat,ateachpointxin(-1 , l), Q,(x) has the same sign as P,(x). (c) Use part (b), along with Exercise 13, to show that the inequality m < n leads to a con- tradiction. This shows that P, has n distinct real zeros, all of which lie in the open interval C-1 , 1).15. (a) Show that the value of the integral j?, P,(x)Ph+,(x) dx is independent of n. (b) Evaluate the integral JL1 x P,(x)PGl(x) dx.6.22 The method of Frobenius In Section 6.17 we learned how to find power-series solutions of the differential equation(6.43) y" + P1(x>y' + P,(x)y = 0in an interval about a point x,, where the coefficients P, and P2 are analytic. If either PI orP2 is not analytic near x,, , power-series solutions valid near x0 may or may not exist. Forexample, suppose we try to find a power-series solution of the differential equation(6.44) x2y" -y-y=onear x0 = 0. If we assume that a solution y = 2 akxk exists and substitute this series in thedifferential equation we are led to the recursion formula n2-n-l a n+1 = a,. n+l
The method of Frobenius ia1Although this gives us a power series y = 2 akxk which formally satisfies (6.44), the ratiotest shows that this power series converges only for x = 0. Thus, there is no power-seriessolution of (6.44) valid in any open interval about x,, := 0. This example does not violateTheorem 6.13 because when we put Equation (6.44) in the form (6.43) we find that thecoefficients PI and P, are given by PI(X) = - 1 and P,(x) = - L . x2 X2These functions do not have power-series expansions about the origin. The difficulty hereis that the coefficient of y" in (6.44) has the value 0 when x = 0 ; in other words, thedifferential equation has a singular point at x = 0. A knowledge of the theory of functions of a complex variable is needed to appreciate thedifficulties encountered in the investigation of differential equations near a singular point.However, some important special cases of equations with singular points can be treated byelementary methods. For example, suppose the differential equation in (6.43) is equivalentto an equation of the form(6.45) (x - x,)2y" + (x - x,)P(x)y' + ec.x>v = 0 9where P and Q have power-series expansions in some open interval (x,, - r, x0 + r). Inthis case we say that x0 is a regular singular point of the equation. If we divide both sidesof (6.45) by (x - x0)2 the equation becomes Y" + - J'(x) y, + Q(x) x - x0 ( x - x0)2 y = Ofor x # x0. If P(x,,) # 0 or Q(xJ # 0, or if Q(x,,) = 0 and Q'(x,) # 0, either the co-efficient ofy' or the coefficient ofy will not have a power-series expansion about the point x0,so Theorem 6.13 will not be applicable. In 1873 the German mathematician Georg Fro-benius (1849-1917) developed a useful method for treating such equations. We shalldescribe the theorem of Frobenius but we shall not present its proof.? In the next sectionwe give the details of the proof for an important special case, the Bessel equation. Frobenius' theorem splits into two parts, depending on the nature of the roots of thequadratic equation(6.46) t(t - 1) + P(x& + Q(xo> = 0.This quadratic equation is called the indicial equation of the given differential equation(6.45). The coefficients P(x,J and Q(x,) are the constant terms in the power-series ex-pansions of P and Q. Let tcl and x2 denote the roots of the indicial equation, These rootsmay be real or complex, equal or distinct. The type of solution obtained by the Frobeniusmethod depends on whether or not these roots differ by an integer.t For a proof see E. Hille, Analysis, Vol. II, Blaisdell Publishing Co., 1966, or E. A. Coddington, AnIntroduction to Ordinary Dijkrential Equations, Prentice-Hall, 1961.
182 Linear d@erentiaI equations THEOREM 6.14. FIRST CASE OF FROBENIUS THEOREM. Let u1 and u2 be the roots of theindicial equation and assume that uI - u2 is not an integer. Then the difSerentia1 equation(6.45) has two independent solutions u1 and u2 of the form(6.47) with a, = 1,and uz(x) = Ix - x01+ b,(x - xoy, with b, = 1. ?L=OBoth series converge in the interval Ix - x01 < r , and the differential equation is satisfied forO<Jx-x,J<r. THEOREM 6.15. SECOND CASE OF FROBENIUS THEOREM. Let ul, u2 be the roots ofthe indicial equation and assume that CQ - u2 = N, a nonnegative integer. Then the d@erential equation (6.45) has a solution u1 of the form (6.47) and another independent solution uZ of theform(6.49) u2(x) = lx - xolaz~ b,(x - x~)~ + C ul(x) log 1.x - x01, n=Owhere b, = 1 . The constant C is nonzero if N = 0. If N > 0, the constant C may or maynot be zero. As in Case 1, both series converge in the interval Ix - x01 < r , and the solutionsare validfor 0 < Ix - x0( < r .6.23 The Bessel equation In this section we use the method suggested by Frobenius to solve the Bessel equation x2y" + xy' + (x2 - u")y = 0 ,where 0: is a nonnegative constant. This equation is used in problems concerning vibrationsof membranes, heat flow in cylinders, and propagation of electric currents in cylindricalconductors. Some of its solutions are known as Besselfunctions. Bessel functions also arisein certain problems in Analytic Number Theory. The equation is named after the Germanastronomer F. W. Bessel (1784-1846), although it appeared earlier in the researches ofDaniel Bernoulli (1732) and Euler (1764). The Bessel equation has the form (6.45) with x0 = 0, P(X) = 1, and Q(X) = x2 - u2,so the point x0 is a regular singular point. Since P and Q are analytic on the entire real line,we try to find solutions of the form(6.50) y = Ixltza,xn, 71=0with a, # 0, valid for all real x with the possible exception of x = 0. First we keep x > 0, so that lxlt = xt . Differentiation of (6.50) gives us y' = txt-1 5 a,xn + xt 2 na,xn-' = xt--l ZJn + W$. ?l=O TL=O
184 Linear d@erential equations In this discussion we assumed that x > 0. If x < 0 we can repeat the discussion withxt replaced by (-x)" . We again find that t must satisfy the equation t2 - c? = 0. Takingt = u we then obtain the same solution, except that the outside factor xa is replaced by(-x)". Therefore the functionf, given by the equation(6.54) fAx) = ao IxIa is a solution of the Bessel equation valid for all real x # 0. For those values of cc for whichf:(O) andfl(O) exist the solution is also valid for x = 0. Now consider the root t = -LX of the indicial equation. We obtain, in place of (6.52), the equations [(l - cc)" - c?]al = 0 and Kn - 4" - cr2]a, + an-2 = 0,which become (1 - 20()a, = 0 and 0 - 2a)a, + ane2 = 0.If 2u is not an integer these equations give us a, = 0 and anm2 a, = - n(n - 2cc)for n 2 2. Since this recursion formula is the same as (6.53), with CI replaced by -CI, weare led to the solution(6.55) f-Ax) = ao W" l + 2& 22nn( (1 _ &y;. . . (n _ u)valid for all real x # 0. The solution f-, was obtained under the hypothesis that 2cr is not a positive integer.However, the series forf_, is meaningful even if 2u is a positive integer, so long as CC isnot a positive integer. It can be verified thatf-, satisfies the Bessel equation for such CC.Therefore, for each M 2 0 we have the series solution f,, given by Equation (6.54); andif cI is not a nonnegative integer we have found another solutionf-, given by Equation (6.55).The two solutions f, and f-, are independent, since one of them --to3 as x + 0, and theother does not. Next we shall simplify the form of the solutions. To do this we need someproperties of Euler's gamma function, and we digress briefly to recall these properties. For each real s > 0 we define I'(s) by the improper integral l?(s) = lo: ts-le-t dt .
186 Linear difSerentia1 equationsThe function J, defined by this equation for x > 0 and M 2 0 is called the Bessel/unctionof thejrst kind of order u. When u is a nonnegative integer, say CI = p , the Bessel functionJ, is given by the power series J,(x) = z. ., :;yp,, (5"'" (p = 0, 1,2, . . .).This is also a solution of the Bessel equation for x < 0. Extensive tables of Bessel functionshave been constructed. The graphs of the two functions Jo and J1 are shown in Figure 6.2. t Jo FIGURE 6.2 Graphs of the Bessel functions Jo and J1. We can define a new function I, by replacing CI by -u in Equation (6.59), if u is suchthat P(n + 1 - u) is meaningful; that is, if cc is not a positive integer. Therefore, if x > 0andu>O,u# 1,2,3 ,..., wedeline J-,(x) = 0-& r(;--'; _ .,($'. 5Taking s = 1 - u in (6.57) we obtain r(n + 1 - LX) = (1 - ct) (2 - U) . . . (n - U) I?( 1 - U)and we see that the series for J-,(x) is the same as that for f-,(x) in Equation (6.55) witha, = 2"/I'(l - u), x > 0. Therefore, if u is not a positive integer, J-, is a solution ofthe Bessel equation for x > 0. If u is not an integer, the two solutions J,(x) and J-,(x) are linearly independent on thepositive real axis (since their ratio is not constant) and the general solution of the Besselequation for x > 0 is y = c,J,(x) + c,J&) . If u is a nonnegative integer, say cx = p , we have found only the solution J, and its con-stant multiples valid for x > 0. Another solution, independent of this one, can be found
The Bessel equation 187by the method described in Exercise 4 of Section 6.16. This states that if u1 is a solution ofy" + Ply' + P,y = 0 that never vanishes on an interval I, a second solution u2 independentof u1 is given by the integralwhere Q(x) = e-J1'l(s)dr. For the Bessel equation we have PI(x) = l/x, so Q(x) = l/x up(x) = J,(x) s "1dt*and a second solution u2 is given by the formula(6.60) c w&vif c and x lie in an interval I in which J, does not vanish. This second solution can be put in other forms. For example, from Equation (6.59) wemay write 1 = r s,(t), [4(01" tznwhere g,(O) # 0. In the interval Z the function g, has a power-series expansion s,(t) = 2 Ant" VI=0which could be determined by equating coefficients in the identity g,(t) [J,(t)]2 = Pp. Ifwe assume the existence of such an expansion, the integrand in (6.60) takes the form 1 1 m c Ant". f[J,(f>]" = t2"+1 n=OIntegrating this formula term by term from c to x we obtain a logarithmic term A,, log x(from the power t-l) plus a series of the form xP2P 2 B,?. Therefore Equation (6.60)takes the form u2(x) = A2,J,(x) log x + Jy(x)x-2D~ B,x" . ?k=OIt can be shown that the coefficient A,, # 0. If we multiply u2(x) by l/A,, the resultingsolution is denoted by K,(x) and has the form K,(x) = J,(x) log x + x-p2 c,,xn. 7L=OThis is the form of the solution promised by the second case of Frobenius' theorem. Having arrived at this formula, we can verify that a solution of this form actually existsby substituting the right-hand member in the Bessel equation and determining the co-efficients C, so as to satisfy the equation. The details of this calculation are lengthy and
188 Linear dxerential equationswill be omitted. The final result can be expressed aswhere ho = 0 and h, = 1 + 4 + * * * + l/n for n 2 1. The series on the right convergesfor all real x. The function K, defined for x > 0 by this formula is called the Besselfunctionof the second kind of orderp. Since K, is not a constant multiple of J, , the general solutionof the Bessel equation in this case for x > 0 is Y = c,J,(x) + c,K&).Further properties of the Bessel functions are discussed in the next set of exercises.6.24 Exercises 1. (a) Let f be any solution of the Bessel equation of order a and let g(x) = x'hf (x) for x > 0. Show that g satisfies the differential equation 1 -44a2 ( y" + 1 + 4x2 y = o . 1 (b) When 4a2 = 1 the differential equation in (a) becomes y" + y = 0 ; its general solution is y = A cos x + B sin x. Use this information and the equation? I'(h) = d, to show that, forx >O, and (c) Deduce the formulas in part (b) directly from the series for Jim(x) and Il,h(x). 2. Use the series representation for Bessel functions to show that (a) 2 (x~J,W> = x~J,-~W, (b) $ (x-='J,(x)) = -xP-r~+,(x) 3. Let F,(x) = x"J,(x) and 'G,(x) = xaaJ,(x) for x > 0. Note that each positive zero of J, is a zero of F, and is also a zero of G, . Use Rolle's theorem and Exercise 2 to prove that the posi- tive zeros of J, and J,+r interlace. That is, there is a zero of J, between each pair of positive zeros of Ja+l, and a zero of J,+l between each pair of positive zeros of J,. (See Figure 6.2.)t The change of variable I = u* gives us r(t) = Jo; t-/&-t dt = 2 s ")e-d du = 4,. 0(See Exercise 16 of Section 11.28 for a proof that 2 Jo" e-u2 du = 2/G.)
190 Linear difSerentia1 equations12. Find a power series solution of the differential equation xy" + y' + y = 0 convergent for - - t o < x < +a. Show that for x > 0 it can be expressed in terms of a Bessel function.13. Consider a linear second-order differential equation of the form X2&)yn + xP(x)y' + Q(x)y = 0, where A(x), P(x), and Q(x) have power series expansions, A(x) = 2 UkXk) P(x) = ygPkXk, k=O k=O with a, # 0, each convergent in an open interval (-r, r) . If the differential equation has a series solution of the form y = xt 2 CnXll ) ?I=0 valid for 0 < x < Y, show that t satisfies a quadratic equation of the form t2 + bt + c = 0, and determine b and c in terms of coefficients of the series for A(x), P(x), and Q(x).14. Consider a special case of Exercise 13 in which A(x) = 1 - x, P(X) = $, and Q(x) = -8~. Find a series solution with t not an integer.15. The differential equation 2x2y" + (x2 - x)y' + y = 0 has two independent solutions of the form y =Xt~C,,x", T&=0 valid for x > 0. Determine these solutions.16. The nonlinear differential equation y" + y + my 2 = 0 is only "mildly" nonlinear if c( is a small nonzero constant. Assume there is a solution which can be expressed as a power series in a of the form y = g U,(X)@ (valid in some interval 0 < CL < r) ?I=0 and that this solution satisfies the initial conditions y = 1 and y' = 0 when x = 0. To con- form with these initial conditions, we try to choose the coefficients U,(X)SO that U,(O) = 1, u;(O) = 0 and u,(O) = u;(O) = 0 for n 2 1 . Substitute this series in the differential equation, equate suitable powers of c( and thereby determine uo(x) and ut(x).
7 SYSTEMS OF DIFFERENTIAL EQUATIONS7.1 Introduction Although the study of differential equations began in the 17th century, it was not untilthe 19th century that mathematicians realized that relatively few differential equations couldbe solved by elementary means. The work of Cauchy, Liouville, and others showed theimportance of establishing general theorems to guarantee the existence of solutions tocertain specific classes of differential equations. Chapter 6 illustrated the use of an existence-uniqueness theorem in the study of linear differential equations. This chapter is concernedwith a proof of this theorem and related topics. Existence theory for differential equations of higher order can be reduced to the first-order case by the introduction of systems of equations. For example, the second-orderequation(7.1) y" + 22~' - y = etcan be transformed to a system of two first-order equations by introducing two unknownfunctions y, and yZ , where Yl=YY Y2 = YI.Then we have y6 = yl = y", so (7.1) can be written as a system of two first-order equations : Y; = Y2(7.2) yJ = y, - 2ty2 + et.We cannot solve the equations separately by the methods of Chapter 6 because each of theminvolves two unknown functions. In this chapter we consider systems consisting of n linear differential equations of firstorder involving n unknown functions yI, . . . , y, . These systems have the form YI = Pll(OY1 + P12COY2 + * * . + Pln(OYn + q1(0(7.3) Y:, = I&JOY1 + P,2(9Y2 + * * * + Pm(9Yn + q,(t). 191
192 Systems of deferential equationsThe functionsp, and qi which appear in (7.3) are considered as given functions defined ona given interval J. The functions yr, . . . , yn are unknown functions to be determined.Systems of this type are called$rst-order linear systems. In general, each equation in thesystem involves more than one unknown function so the equations cannot be solvedseparately. A linear differential equation of order n can always be transformed to a linear system.Suppose the given nth order equation is(7.4) ytn) + aly@-l) + * - * + a,y = R(t),where the coefficients ai are given functions. To transform this to a system we write y1 = yand introduce a new unknown function for each of the successive derivatives of y. That is,we put Yl=Y, Y2 = Y;, Y3 = Yi!, . . . 3 Yn = YL,and rewrite (7.4) as the system YI = Y2 Y;; = Y3(7.5) I Yn-1 = Yn y; = -any1 - a,-,y2 - * * * - alyn + R(t). The discussion of systems may be simplified considerably by the use of vector and matrixnotation. Consider the general system (7.3) and introduce vector-valued functions Y =cY17.. . ,Y,), Q = (41,. . . ,q,J, and a matrix-valued function P = [pij], defined by theequations y(t) = O1(t>, * * * 9 y,(t)>, Q(t) = @l(t), . . . 9 qn(t>>, PC0 = h(t)1for each t in J. We regard the vectors as n x 1 column matrices and write the system (7.3)in the simpler form(7.6) Y' = P(t) Y + Q(t). For example, in system (7.2) we have Y= K;], P(t)= [; _:,I, Q(t)= [;j.
Calculus of matrix functions 193In system (7.5) we have -0 1 0 . . . 0- 'o- r1- 0 0 1 . . . 0 0 Y2 Y= 2 P(t) = , Q(t) = . 0 0 0 . . . 1 0 Yn- ---a, -a,-, -an-2 * ** - a , - .R(t)- An initial-value problem for system (7.6) is to find a vector-valued function Y whichsatisfies (7.6) and which also satisfies an initial condition of the form Y(a) = B, whereaEJandB=(b,,.. . , b,) is a given n-dimensional vector. In the case n = 1 (the scalar case) we know from Theorem 6.1 that, if P and Q arecontinuous on J, all solutions of (7.6) are given by the explicit formula(7.7) Y(X) = &(')Y(a) + eA(') s ," e-act)Q(f) dt,where A(x) = j; P(t) dt , and a is any point in J. We will show that this formula can besuitably generalized for systems, that is, when P(t) is an n x n matrix function and Q(t)is an n-dimensional vector function. To do this we must assign a meaning to integrals ofmatrices and to exponentials of matrices. Therefore, we digress briefly to discuss thecalculus of matrix functions.7.2 Calculus of matrix functions The generalization of the concepts of integral and derivative for matrix functions isstraightforward. If P(t) = [pi,(t)], we define the integral jz P(t) dt by the equationThat is, the integral of matrix P(t) is the matrix obtained by integrating each entry of P(t),assuming of course, that each entry is integrable on [a, b]. The reader can verify that thelinearity property for integrals generalizes to matrix functions. Continuity and differentiability of matrix functions are also defined in terms of theentries. We say that a matrix function P = [pij] is continuous at t if each entry pij iscontinuous at t. The derivative P' is defined by differentiating each entry,whenever all derivatives pi,(t) exist. It is easy to verify the basic differentiation rules forsums and products. For example, if P and Q are differentiable matrix functions, we have (P + Q)' = P' + Q'
194 Systems of d@erential equationsif P and Q are of the same size, and we also have (PQ)' = PQ' + P'Qif the product PQ is defined. The chain rule also holds. That is, if F(t) = P[g(t)] , where Pis a differentiable matrix function and g is a differentiable scalar function, then F'(t) =g'(t)P'[g(t)] . The zero-derivative theorem, and the first and second fundamental theoremsof calculus are also valid for matrix functions. Proofs of these properties are requested inthe next set of exercises. The definition of the exponential of a matrix is not so simple and requires furtherpreparation. This is discussed in the next section.7.3 Infinite series of matrices. Norms of matrices Let A = [aij] be an n x n matrix of real or complex entries. We wish to define theexponential en in such a way that it possesses some of the fundamental properties of theordinary real or complex-valued exponential. In particular, we shall require the law ofexponents in the form(7.8) for all real s and t ,and the relation(7.9) e"=I,where 0 and Z are the n x n zero and identity matrices, respectively. It might seem naturalto define eA to be the matrix [e@j]. However, this is unacceptable since it satisfies neither ofproperties (7.8) or (7.9). Instead, we shall define eA by means of a power series expansion,We know that this formula holds if A is a real or complex number, and we will prove thatit implies properties (7.8) and (7.9) if A is a matrix. Before we can do this we need toexplain what is meant by a convergent series of matrices. DEFINITION OF CONVERGENT SERIES OF MATRICES. Given an infkite sequence of m X nmatrices {Ck} whose entries are real or complex numbers, denote the ij-entry of C, by cl,"' . Zfall mn series(7.10) $6' (i=l,..., m;j=l,..., n)are convergent, then we say the series of matrices z& C, is convergent, and its sum is deJnedto be the m x n matrix whose ij-entry is the series in (7.10).
Exercises 195 A simple and useful test for convergence of a series of matrices can be given in terms of thenorm of a matrix, a generalization of the absolute value of a number. DEFINITION OF NORM OF A MATRIX. If A = [aij] is an m x n matrix of real or complex entries, the norm of A, denoted by lAll, is defined to be the nonnegative number given by theformula(7.11) In other words, the norm of A is the sum of the absolute values of all its entries. Thereare other definitions of norms that are sometimes used, but we have chosen this one becauseof the ease with which we can prove the following properties. THEOREM 7.1. FUNDAMENTAL PROPERTIES OF NORMS. For rectangular matrices A and B,and all real or complex scalars c we have II-4 + AI I IIAII + IIBII, IWll I II4 IIBII 3 II4 = ICI IIAII . Proof. We prove only the result for I ABll , assuming that A is m x n and B is n x p .The proofs of the others are simpler and are left as exercises. Writing A = [aJ , B = [bkj] , we have AB = [z;==, ai,bkj] , so from (7.11) we obtain IIABII = i=l I$ 1~airhj 1I$ ~laikl~lhjl 52 ~I4 IIBII = IIAII II4 . 2 j,l J+l i=lk=l j=l i=l k=l Note that in the special case B = A the inequality for [lABI/ becomes llA211 5 llAl12.By induction we also have llA"II I II4" for k=l,2,3 ,....These inequalities will be useful in the discussion of the exponential matrix. The next theorem gives a useful sufficient condition for convergence of a series of matrices. THEOREM 7.2. TEST FOR CONVERGENCE OF A MATRIX SERIES. If {c,} is a sequence Of Wl X nmatrices such that I;=1 I C,ll converges, then the matrix series zF-'em=l C, also converges. Proof. Let the q-entry of C, be denoted by c$'. Since lcj,k'l 5 IlC,(l , convergence of121 Ilckll i mpl' ies absolute convergence of each series xTZl cjj"'. Hence each series 2:-l c$'is convergent, so the matrix series z;Cmcl C, is convergent.7.4 Exercises 1. Verify that the linearity property of integrals also holds for integrals of matrix functions. 2. Verify each of the following differentiation rules for matrix functions, assuming P and Q are differentiable. In (a), P and Q must be of the same size so that P + Q is meaningful. In
196 Systems of d@erential equations (b) and (d) they need not be of the same size provided the products are meaningful. In (c) and (d), Q is assumed to be nonsingular. (a) (P + Q)' = P' + Q' . (c) (Q-l)' = -Q-lQ'Q-l. (b) (PQ)' = PQ' + P'Q. (d) (PQ-')' = -PQ-lQ'Q-l + P'Q-l.3. (a) Let P be a differentiable matrix function. Prove that the derivatives of P2 and P3 are given by the formulas (P2)' = PP' + P'P, (P3)' = P2P' + PP'P + P'P2. (b) Guess a general formula for the derivative of Pk and prove it by induction.4. Let P be a differentiable matrix function and let g be a differentiable scalar function whose range is a subset of the domain of P. Define the composite function F(t) = P[g(t)] and prove the chain rule, F'(t) =g'(t)P'[g(t)J .5. Prove the zero-derivative theorem for matrix functions: If P'(t) = 0 for every t in an open interval (a, b), then the matrix function P is constant on (a, b).6. State and prove generalizations of the first and second fundamental theorems of calculus for matrix functions.7. State and prove a formula for integration by parts in which the integrands are matrix functions.8. Prove the following properties of matrix norms : IM + WI I IIA II + IIBII , IlcAII = ICI IMII . 9. If a matrix function P is integrable on an interval [a, b] prove that10. Let D be an n x n diagonal matrix, say D = diag (Al, . . . , A,). Prove that the matrix series z:km_s D"/k! converges and is also a diagonal matrix, c m D" k=O - = diag (ear, . . . , e"n), k! (The term corresponding to k = 0 is understood to be the identity matrix I.)11. Let D be an n x n diagonal matrix, D = diag (jlr , . . . , I,). If the matrix series I:=, ckDk converges, prove that 2 CkDk = diag 2 c&, . . * z. c&) * k=O k=O12. Assume that the matrix series IF=1 ck converges, where each ck is an n x n matrix. Prove that the matrix series I:& (AC&I) also converges and that its sum is the matrix Here A and B are matrices such that the products ACkB are meaningful.
The diferential equation satisjed by et' 1977.5 The exponential matrix Using Theorem 7.2 it is easy to prove that the matrix series(7.12) cm Ak kc0 72converges for every square matrix A with real or complex entries. (The term correspondingto k = 0 is understood to be the identity matrix Z.) The norm of each term satisfies theinequalitySince the series 2 a"/k! converges for every real a, Theorem 7.2 implies that the series in(7.12) converges for every square matrix A. DEFINITION OF THE EXPONENTIAL MATRIX. For any n x n matrix A with real or complexentries we dejne the exponential & to be the n x n matrix given by the convergent series in(7.12). That is, Note that this definition implies e* = I, where 0 is the zero matrix. Further propertiesof the exponential will be developed with the help of differential equations.7.6 The differential equation satisfied by et* Let t be a real number, let A be an n x n matrix, and let E(t) be the n x n matrix given by E(t) = d" .We shall keep A fixed and study this matrix as a function oft. First we obtain a differentialequation satisfied by E. T H E O R E M 7.3. For every real t the matrix function E defined by E(t) = etA satisfies thematrix diferential equation E'(t) = E(t)A = AE(t). Proof. From the definition of the exponential matrix we have
198 Systems of difSerentia1 equationsLet ci,"' denote the ij-entry of Ak. Then the ij-entry of tkAklk! is t"c$'lk! . Hence, from thedefinition of a matrix series, we have(7.13)Each entry on the right of (7.13) is a power series in t, convergent for all t. Therefore itsderivative exists for all t and is given by the differentiated series c- m ktk-1 $9 = k=l k! "This shows that the derivative E'(t) exists and is given by the matrix series A = lqt)A.In the last equation we used the property Ak+l = A"A . Since A commutes with A' we couldalso have written A"+l = AAL to obtain the relation E'(t) = AE(t). This completes theproof. Note: The foregoing proof also shows that A commutes with etA .7.7 Uniqueness theorem for the matrix differential equation F'(t) = M(t) In this section we prove a uniqueness theorem which characterizes all solutions of thematrix differential equation F'(t) = AF(t) . The proof makes use of the following theorem. THEOREM 7.4. NONSINGULARlTY OF etA. For any n x n matrix A and any scalar t we have(7.14)Hence etA is nonsingular, and its inverse is e-tA. Proof. Let F be the matrix function defined for all real t by the equation F(t) = etAeetA.We shall prove that F(t) is the identity matrix I by showing that the derivative F'(t) is thezero matrix. Differentiating F as a product, using the result of Theorem 7.3, we find F'(t) = etA(e--tA)f + (etA>'e-tA = &4(-A&A) + A&f+' = -AetAe-tA + AetAe-tA = 0,since A commutes with etA. Therefore, by the zero-derivative theorem, F is a constantmatrix. But F(0) = eoAeoA = I, so F(t) = I for all t. This proves (7.14).
The law of exponents for exponential matrices 199 THEOREM 7.5. UNIQUENESS THEOREM. Let A and B be given n x n constant matrices.Then the only n x n matrix function F satisfying the initial-value problem F'(t) = AF(t), F(0) = Bfor - a3 < t < + 00 is(7.15) F(t) = etaB . Proof. First we note that etAB is a solution. Now let F be any solution and consider thematrix function G(t) = eWtaF(t) .Differentiating this product we obtain G'(t) = ePtAF'(t) - AePtaF(t) = ePtAAF(t) - eTtAAF(t) = 0.Therefore G(t) is a constant matrix, G(t) = G(0) = F(0) = B.In other words, e-taF(t) = B. Multiplying by eta and using (7.14) we obtain (7.15). Note: The same type of proof shows that F(t) = BetA is the only solution of the initial-value problem F(t) = FQ)A, F(0) = B.7.8 The law of exponents for exponential matrices The law of exponents eAeB = eAfB is not always true for matrix exponentials. A counterexample is given in Exercise 13 of Section 7.12. However, it is not difficult to prove thatthe formula is true for matrices A and B which commute. T HEORE M 7.6. Let A and B be two n x n matrices which commute, AB = BA . Then wehave(7.16) I++~ = eAeB. Proof. From the equation AB = BA we find that A2B = A(BA) = (AB)A = (BA)A = BA2,so B commutes with A2. By induction, B commutes with every power of A. By writingetA as a power series we find that B also commutes with eta for every real t. Now let F be the matrix function defined by the equation I = $(A+B) - etAetBe
200 Systems of differential equationsDifferentiating F(t) and using the fact that B commutes with eta we find F'(t) = (A + B)et'A+B' - AetAetB - etABetB = (A + B)et'A+B' - (A + B)etAetB = (A + B)F(t) .By the uniqueness theorem we have F(t) = et(d+B)F(0).But F(0) = 0, so F(t) = 0 for all t. Hence $(A+B) = etAetBWhen t = 1 we obtain (7.16). EXAMPLE. The matrices SA and tA commute for all scalars s and t. Hence we have @etA = e (s+t)A .7.9 Existence and uniqueness theorems for homogeneous linear systems with constant coefficients The vector differential equation Y'(t) = A Y(t), where A is an n x n constant matrix and Y is an n-dimensional vector function (regarded as an n x 1 column matrix) is called ahomogeneous linear system with constant coeficients. We shall use the exponential matrixto give an explicit formula for the solution of such a system. THEOREM 7.7. Let A be a given n x n constant matrix and let B be a given n-dimensionalvector. Then the initial-value problem(7.17) Y'(t) = AY(t), Y(0) = B,has a unique solution on the interval - CO < t < + CO. This solution is given by the formula(7.18) Y(t) = etAB.More generally, the unique solution of the initial value problem Y'(t) = A Y(t), Y(a) = B,is Y(t) = ectaJAB. Proof. Differentiation of (7.18) gives us Y'(t) = AetAB = A Y(t). Since Y(0) = B ,this is a solution of the initial-value problem (7.17). To prove that it is the only solution we argue as in the proof of Theorem 7.5. Let Z(t)be another vector function satisfying Z'(t) = AZ(t) with Z(0) = B , and let G(t) = emtAZ(t).Then we easily verify that G'(t) = 0, so G(t) = G(0) = Z(0) = B. In other words,
The problem of calculating et' 201etAZ(t) = B, so Z(t) = etAB = Y(t). The more general case with initial value Y(a) = Bis treated in exactly the same way.7.10 The problem of calculating erA Although Theorem 7.7 gives an explicit formula for the solution of a homogeneoussystem with constant coefficients, there still remains the problem of actually computingthe exponential matrix etA. If we were to calculate etA directly from the series definition wewould have to compute all the powers AL for k = 0, 1,2, , . . , and then compute the sumof each series z& tkc$)/k! , where c:j"' is the q-entry of Ak. In general this is a hopelesstask unless A is a matrix whose powers may be readily calculated. For example, if A is adiagonal matrix, say A = diag (1,) . . . , 1,))then every power of A is also a diagonal matrix, in fact, A'=diag(;I:,...,iZE).Therefore in this case etA is a diagonal matrix given by eiA = diag = diag (et"', . . . , et""). Another easy case to handle is when A is a matrix which can be diagonalized. Forexample, if there is a nonsingular matrix C such that PAC is a diagonal matrix, sayC-lAC = D , then we have A = CDC-l , from which we find A2 = (CDC-l)(CDC-l) = CDW-l,and, more generally, Ak = CD"C-I.Therefore in this case we have etA = 2 $ik= 2 +; CDkC-1 = c k=O * k=O 'Here the difficulty lies in determining C and its inverse. Once these are known, etA is easilycalculated. Of course, not every matrix can be diagonalized so the usefulness of theforegoing remarks is limited. [1 5 4 EXAMPLE 1. Calculate etA for the 2 x 2 matrix A = 1 2 ' Solution. This matrix has distinct eigenvalues 1, = 6, 1, = 1 , so there is a nonsingularmatrix C = such that C-lAC = D, where D = diag (&, 1,) = To
202 Systems of deferential equationsdetermine C we can write AC = CD, orMultiplying the matrices, we find that this equation is satisfied for any scalars a, b, c, d witha=4c,b= - d . Takingc=d=lwechooseTherefore etA EXAMPLE 2. Solve the linear system YI = 5Yl + 4Y, Y;; = Yl + 2Y,subject to the initial conditions ~~(0) = 2, ~~(0) = 3. Solution. In matrix form the system can be written as Y'(t) = A Y(t), where A =By Theorem 7.7 the solution is Y(t) = etA Y(0). Using the matrix etA calculated in Example1 we find I[1 4eet + et 4e6t - 4et 2 e6t - et est + 4et 3from which we obtain yl = 4e6t - 2et, yz = eat + 2et. There are many methods known for calculating etA when A cannot be diagonalized.Most of these methods are rather complicated and require preliminary matrix transforma-tions, the nature of which depends on the multiplicities of the eigenvalues of A. In a latersection we shall discuss a practical and straightforward method for calculating etA whichcan be used whether or not A can be diagonalized. It is valid for all matrices A and requiresno preliminary transformations of any kind. This method was developed by E. J. Putzerin a paper in the American Mathematical Monthly, Vol. 73 (1966), pp. 2-7. It is based on afamous theorem attributed to Arthur Cayley (1821-1895) and William Rowan Hamilton
The Cayley-Hamilton theorem 203(18051865) which states that every square matrix satisfies its characteristic equation.First we shall prove the Cayley-Hamilton theorem and then we shall use it to obtainPutzer's formulas for calculating etA.7.11 The Cayley-Hamilton theorem THEOREM 7.8. CAYLEY-HAMILTON THEOREM. Let A be an n x n matrix and let(7.19) f(a) = det (az - A) = an + C,-lan--l + . e. + c,a + C,be its characteristic polynomial. Then f (A) = 0. In other words, A satisfies the equation(7.20) A" + c,-~A'@ + . * * + c,A + co1 = 0. Proof. The proof is based on Theorem 3.12 which states that for any square matrix Awe have(7.21) A (cof A)t = (det A)I.We apply this formula with A replaced by ;ZI - A . Since det (AZ - A) = f(a), Equation(7.21) becomes(7.22) (a1 - A)(cof (al - ~))t =f(w.This equation is valid for all real 1. The idea of the proof is to show that it is also validwhen ;3. is replaced by A. The entries of the matrix cof (AZ - A) are the cofactors of LI - A. Except for a factorf 1 , each such cofactor is the determinant of a minor of 2I- A of order n - 1. Thereforeeach entry of cof (U - A), and hence of (cof (U - A)}*, is a polynomial in j2 of degree<_n - 1. Therefore n-1 +0f (ar - ~)y = zakBk, k=Owhere each coefficient Bk is an n x n matrix with scalar entries. Using this in (7.22) weobtain the relation 11-l(7.23) (aI - A)Z:~~B, =f(a)z k=Owhich can be rewritten in the form n-1 S-l(7.24) a"&-, + k~lak@L-l - A&) - AB, = a"1 + 2 akckl i- CoI. k=l
204 Systems of diflerential equationsAt this stage we equate coefficients of like powers of 3, in (7.24) to obtain the equations B,-, = I &-a - AB,-l = c,-J .(7.25) B, - AB, = c,Z -AB, = c,Z.Equating coefficients is permissible because (7.24) is equivalent to n2 scalar equations, ineach of which we may equate coefficients of like powers of il. Now we multiply the equa-tions in (7.25) in succession by A", An-l, . . . , A, I and add the results. The terms on theleft cancel and we obtain 0 = A" + c,elAn-l + . . * + c,A + c,,I.This proves the Cayley-Hamilton theorem. Note: Hamilton proved the theorem in 1853 for a special class of matrices. A few years later, Cayley announced that the theorem is true for all matrices, but gave no proof. EXAMPLE. The matrix A = has characteristic polynomial f(l) = (3, - l)(L - 2)(1 - 6) = L3 - 912 + 201 - 12.The Cayley-Hamilton theorem states that A satisfies the equation(7.26) A3-9A2+20A- 12I=O.This equation can be used to express A3 and all higher powers of A in terms of I, A, andAZ. For example, we have A3 = 9A2 - 20A + 121, A4 = 9A3 - 20A2 + 12A = 9(9A2 - 20A + 121) - 20A2 + 12A = 61A2 - 168A + 1081.It can also be used to express A-l as a polynomial in A. From (7.26) we writeA(A2 - 9A + 201) = 121, and we obtain A-1 = ,+&2 - 9A + 201).
Putzer's methodfor calculating etA 2057.12 Exercises In each of Exercises 1 through 4, (a) express A- l , A2 and all higher powers of A as a linearcombination of Z and A. (The Cayley-Hamilton theorem can be of help.) (b) Calculate et&. 5. (a) If A = , prove that et-4 [ I a b (b) Find a corresponding formula for etA when A = , a, b real. -b a [ I t, t t - l 6. If F(t) = prove that eFtt) = eF(et-l) . A(t) [ 1 I t 0 1 ' 7. If A(t) is a scalar function of the derivative of eA(t) is eA(t)A'(t). Compute the derivative of eAtt)A'(t) eAct) when = and show that the result is not equal to either of the two products 0 0 or A'(t)eAct) . In each of Exercises 8,9, 10, (a) calculate An, and express A3 in terms of I, A, A2. (b) CalculateetA . Lo 0 0-l I , express etA as a linear combination of I, A, A2. x2 xy y212.IfA= 1 2 0 2 1 ,provethateA= [ 2Xy X2+Y2 2xy 1 ,wherex=coshlandy=sinhl. 0 1 0 y2 xy x213. This example shows that the equation eA+B = eA e is not always true for matrix exponentials. B. Compute each of the matrices eAeB , eBeA , eA+B when A = [i I O ] andB = [t -i],and note that the three results are distinct.7.13 Putzer's method for calculating etA The Cayley-Hamilton theorem shows that the nth power of any n x n matrix A can be:xpressed as a linear combination of the lower powers I, A, A2, . . . , An-l. It follows that:ach of the higher powers A"+l, An+2, . . . , can also be expressed as a linear combination of
206 Systems of dzrerential equationsI, A, AZ, . . . , An-l. Therefore, in the infinite series defining etA, each term tkAk/k! withk 2 n is a linear combination of t"I, t"A, tkAz, . . . , t"A'+l. Hence we can expect that elAshould be expressible as a polynomial in A of the form n-1(7.27) e tA = ;oqk(f)Ak )where the scalar coefficients qk(t) depend on t. Putzer developed two useful methods forexpressing etn as a polynomial in A. The next theorem describes the simpler of the two ,methods. THEOREM 7.9. Let A,, . . . , 1, be the eigenvalues of an n x n matrix A, and dejne asequence of polynomials in A as follows:(7.28) PO(A) = 1, P,(A) = fi (A - LO, for k=l,2 ,..., n. TiL=lThen we have n-1(7.29) e tA = ;dt)Pk(A) 2where the scalar coefJicients r,(t), . . . , r,(t) are determined recursively from the system oflinear diflerential equations G(t) = Q,(t) , r,(O) = 1 3(7.30) d+dt> = Ak+lrk+l(t) + rk(t), rk+,(o) = o 9 (k=1,2 ,..., n - l ) . Note: Equation (7.29) does not express E tA directly in powers of A as indicated in (7.271, but as a linear combination of the polynomials P,(A), P,(A), . . . , I',-l(A). These polynomials are easily calculated once the eigenvalues of A are determined. Also the multipliers vi(t), . . . , r,(t) in (7.30) are easily calculated. Although this requires solving a system of linear differential equations, this particular system has a triangular matrix and the solutions can be determined in succession. Proof. Let rl(t), . . . , r,(t) be the scalar functions determined by (7.30) and define amatrix function F by the equation n-l(7.31) Fl(t> = 2 rk+l(t)Pk(A) * k=ONote that F(0) = r,(O)P,(A) = I. We will prove that F(t) = etA by showing that Fsatisfies the same differential equation as etA, namely, F'(t) = AF(t) . Differentiating (7.31) and using the recursion formulas (7.30) we obtain w-1 n-1 F'(t) = 2 rL+l(t)Pk(A) = 2 irkct) + Ak+lrk+l(t))Pk(A) ? k=O k=O
Nonhomogeneous linear systems with constant coe$cients 2137.16 Nonhomogeneous linear systems with constant coefficients We consider next the nonhomogeneous initial-value problem(7.41) Y'(t) = A Y(t) + Q(t), Y(a) = B,on an interval J. Here A is an n x n constant matrix, Q is an n-dimensional vector function(regarded as an n x I column matrix) continuous on J, and a is a given point in J. We canobtain an explicit formula for the solution of this problem by the same process used totreat the scalar case. First we multiply both members of (7.41) by the exponential matrix e-tA and rewrite thedifferential equation in the form(7.42) eMtA{ Y'(t) - AY(t)) = ePtAQ(t).The left member of (7.42) is the derivative of the product ePtAY(t). Therefore, if weintegrate both members of (7.42) from a to x, where x E J, we obtain e-"AY(x) - PAY(a) = s," eetAQ(t) dt .Multiplying by ezA we obtain the explicit formula (7.43) which appears in the followingtheorem. THEOREM 7.13. Let A be an n x n constant matrix and let Q be an n-dimensional vectorfunction continuous on an interval J. Then the initial-value problem Y'(t) = A Y(t) + Q(t), Y(a) = B,has a unique solution on J given by the explicit formula(7.43) Y(x) = eczmajAB + esA s e&Q(t) dt. ,' As in the homogeneous case, the difficulty in applying this formula in practice lies in thecalculation of the exponential matrices. Note that the first term, e(zpa)AL3, is the solution of the homogeneous problem Y'(t) =A Y(t), Y(a) = B. The second term is the solution of the nonhomogeneous problem Y'(t) = A Y(t) + Q(t), Y(a) = 0. We illustrate Theorem 7.13 with an example. EXAMPLE. Solve the initial-value problem Y'(t) = AY(t) + Q(t), Y(0) = B,
Exercises 2157.17 Exercises 1. Let 2 be a solution of the nonhomogeneous system Z'(r) = AZ(r) + Q(r), on an interval J with initial value Z(a). Prove that there is only one solution of the non- homogeneous system Y'(t) = AY(t) + Q(t) on J with initial value Y(a) and that it is given by the formula Y(t) = Z(t) + d-4{ Y(a) - Z(a)}. Special methods are often available for determining a particular solution Z(t) which resembles the given function Q(t). Exercises 2, 3, 5, and 7 indicate such methods for Q(t) = C, Q(r) = &C, Q(t) = tmC, and Q(r) = (cos cct)C + (sin ar)D, where C and D are constant vectors. If the particular solution Z(t) so obtained does not have the required initial value, we modify Z(t) as indicated in Exercise 1 to obtain another solution Y(r) with the required initial value. 2. (a) Let A be a constant n x n matrix, B and C constant n-dimensional vectors. Prove that the solution of the system Y'(f) = AY(t) + c, Y(a) = B, on (- 00, + to) is given by the formula Y(x) = e(=-a)AB + C. (b) If A is nonsingular, show that the integral in part (a) has the value {ecz--a)A - Z}A-l. (c) Compute Y(x) explicitly when A=[-: :I, C=[:], B=[:], a=O. 3. Let A be an n x n constant matrix, let B and C be n-dimensional constant vectors, and let a be a given scalar. (a) Prove that the nonhomogeneous system Z'(t) = AZ(t) + eW has a solution of the form Z(t) = eatB if, and only if, (aZ - A)B = C. (b) If a is not an eigenvalue of A, prove that the vector B can always be chosen so that the system in (a) has a solution of the form Z(t) = eztB . (c) If cc is not an eigenvalue of A, prove that every solution of the system Y'(t) = ,4 Y(i) + eZtC has the form Y(t) = elA( Y(0) - B) + eatB, where B = (al - A)-lC. 4. Use the method suggested by Exercise 3 to find a solution of the nonhomogeneous system Y'(t) = AY(t) + eztC, with A=[: ;I, c=[-;I. Y(@=[;]. |
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This updated and revised edition of David Joyner's entertaining "hands-on" tour of group theory and abstract algebra brings life, levity, and practicality to the topics through mathematical toys.
Joyner uses permutation puzzles such as the Rubik's Cube and its variants, the 15 puzzle, the Rainbow Masterball, Merlin's Machine, the Pyraminx, and the Skewb to explain the basics of introductory algebra and group theory. Subjects covered include the Cayley graphs, symmetries, isomorphisms, wreath products, free groups, and finite fields of group theory, as well as algebraic matrices, combinatorics, and permutations.
Featuring strategies for solving the puzzles and computations illustrated using the SAGE open-source computer algebra system, the second edition of Adventures in Group Theory is perfect for mathematics enthusiasts and for use as a supplementary textbook.
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Based on Saxon's proven methods of incremental development and continual review strategies, the Saxon Advanced Math program builds on intermediate algebraic concepts and trigonometry concepts introduced in Algebra 2 and prepares students for future success in calculus, chemistry, physics and social sciences.
This set includes 31 test forms with full test solutions and answer key with answers to all student textbook problem sets. Also features a recommended test administration schedule.
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Saxon Advanced Math: Homeschool Kit With Test Forms
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John H Saxon Jr
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Edition Number:
2
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Saxon Advanced Math
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Weight:
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A collection of puzzles ranging over geometry, probability, number theory, algebra, calculus, and logic. Hints are provided,...
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A collection of puzzles ranging over geometry, probability, number theory, algebra, calculus, and logic. Hints are provided, along with fully worked solutions, and links to related mathematical topics.
is designed to answer many important questions regarding carrers and professional placement of math majors. It...
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This site is designed to answer many important questions regarding carrers and professional placement of math majors. It contains very useful statistical data on the subject as well as information on different connections between mathematics and real life. The site can be very useful for all kinds of professional orientation sessions for math majors.
This statistical homepage contains educational material of interest to faculty and students studying statistics. It is...
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This statistical homepage contains educational material of interest to faculty and students studying statistics. It is intended to augment the author's textbooks entitled "Statistical Methods for Psychology" and "Fundamental Statistics for the Behavioral Sciences."
This site provides support for fundamental computations in calculus: derivatives, integrals plus some algebra and linear...
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This site provides support for fundamental computations in calculus: derivatives, integrals plus some algebra and linear algebra. The most basic function computes derivatives and integrals. For derivatives, step-by-step solutions are presented and for integrals, a password can be purchased to access step-by-step solutions.
This exercise will help the user understand the logic and procedures of hypothesis testing. To make best use of this...
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This be asked several questions and will be given feedback regarding their answers. Detailed solutions are provided, but users should try to answer the questions on their own before consulting the detailed solutions. The end of the tutorial contains some "thought" questions. |
AP Calculus: Barron's or Princeton Review?
Neither. So long as you still have access to a traditional calculus book, that's your best bet for reviewing. --- "Why bash Jesus? He's born, you get presents, he dies, you get chocolate"~Wolfenstein2000 "He's like a piñata. Hence all the bashing."~LS06
Princeton. --- Not changing this sig until John Cena is off RAW (started January 27, 2008)
Ben KenobiPosted 9/28/2009 1:20:40 PM
[This message was deleted at the request of the original poster]
scannerfishPosted 9/28/2009 6:09:48 PM
If you have a lot of money sitting around, see if Kaplan does a review course a month or so before the exam. --- Yes and the far left also hates flag pins and supports voldemort. hellanicus in reference to a 1337_Vladmir topic
Cauchys InequalityPosted 9/28/2009 10:12:16 PM
Neither. Barron's and Princeton Review are horrible. If College Board has one, go with that, and if not, a Kaplan book would be good. Or, of course, a Calc textbook.
--- "The world is a sphere. That means that the east is connected to the west, and the north is connected to the south." ~Enrique, Skies of Arcadia
zaqwsx99221Posted 9/29/2009 12:25:36 PM
If you have a lot of money sitting around, see if Kaplan does a review course a month or so before the exam.
That would be an UTTER waste of time.
I got the Princeton Review book, and while I'm sure it's fine, I have NEVER OPENED IT.
You WILL NOT need it. So if you're just planning ahead for right before AP tests, don't bother. You won't use it at all.
XKillerPenguinXPosted 9/29/2009 4:08:34 PM
I got Princeton. Every Barron's book I've seen has been dense and superfluous. Princeton seems to be great so far.
From: us38 | #002 Neither. So long as you still have access to a traditional calculus book, that's your best bet for reviewing.
Then that would make calculus special. For every other AP class I've taken, study guides were far superior for test preparation. They teach directly to the test, whereas textbooks go over a broad range of information that you won't necessarily need for it. Of course a textbook is always going to be better for general knowledge, but the TC most likely wants something that's best suited for the test. ---
VirtuousWrathPosted 9/29/2009 11:03:44 PM
Here's the thing. If you only focus on passing the AP test then you'll probably wind up doing less than stellar in more advanced calculus. I knew a couple of guys who got 5's on their exams. I got a 4. All three of us were taking the same vector calculus classes and guess what happened with them? One barely passed and the other had to retake the second half of the course where integration was taught. I however had no trouble passing and did so with B's. I'm not saying that I'm all that smart at math but the AP test is not that important so that's why I would go with the one fellow's suggestion of using the actual calculus book as a study guide. It will be your best bet because its not engineered for you to pass a test, it's designed to actually teach you something which is why you're in school. --- So you're saying you want to be... mentally challenged? - Cro Magnon ...and a box of hair-dye. - Catherine |
This book is written to help parents and their children to get over the fear of mathematics. It is a reference tool that describes time saving techniques, addresses areas of math that students find most difficult, and shares different ways of explaining problems that many students find challenging.
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Chapter 9: Polynomials and Factoring
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PROFESSOR STRANG: Finally
we get to positive
10
00:00:22 --> 00:00:25
definite matrices.
11
00:00:25 --> 00:00:29
I've used the word and now
it's time to pin it down.
12
00:00:29 --> 00:00:33
And so this would be my thank
you for staying with it while
13
00:00:33 --> 00:00:37
we do this important
preliminary stuff
14
00:00:37 --> 00:00:39
about linear algebra.
15
00:00:39 --> 00:00:44
So starting the next lecture
we'll really make a big start
16
00:00:44 --> 00:00:46
on engineering applications.
17
00:00:46 --> 00:00:51
But these matrices are going
to be the key to everything.
18
00:00:51 --> 00:00:58
And I'll call these matrices K
and positive definite, I will
19
00:00:58 --> 00:01:03
only use that word about
a symmetric matrix.
20
00:01:03 --> 00:01:07
So the matrix is already
symmetric and that means it
21
00:01:07 --> 00:01:12
has real eigenvalues and many
other important properties,
22
00:01:12 --> 00:01:14
orthogonal eigenvectors.
23
00:01:14 --> 00:01:17
And now we're asking for more.
24
00:01:17 --> 00:01:26
And it's that extra bit
that is terrific in all
25
00:01:26 --> 00:01:28
kinds of applications.
26
00:01:28 --> 00:01:31
So if I can do this bit
of linear algebra.
27
00:01:31 --> 00:01:34
So what's coming then, my
review session this afternoon
28
00:01:34 --> 00:01:41
at four, I'm very happy that
we've got, I think, the
29
00:01:41 --> 00:01:47
best MATLAB problem ever
invented in 18.085 anyway.
30
00:01:47 --> 00:01:51
So that should get onto the
website probably by tomorrow.
31
00:01:51 --> 00:01:55
And Peter Buchak is like
the MATLAB person.
32
00:01:55 --> 00:01:59
So his review sessions
are Friday at noon.
33
00:01:59 --> 00:02:02
And I just saw him and
suggested Friday at
34
00:02:02 --> 00:02:06
noon he might as well
just stay in here.
35
00:02:06 --> 00:02:10
And knowing that that
isn't maybe a good
36
00:02:10 --> 00:02:11
hour for everybody.
37
00:02:11 --> 00:02:16
So you could see him also
outside of that hour.
38
00:02:16 --> 00:02:20
But that's the hour he will
be ready for all kinds of
39
00:02:20 --> 00:02:23
questions about MATLAB
or about the homeworks.
40
00:02:23 --> 00:02:31
Actually you'll be probably
thinking more also about the
41
00:02:31 --> 00:02:35
homework questions
on this topic.
42
00:02:35 --> 00:02:40
Ready for positive definite?
43
00:02:40 --> 00:02:43
You said yes, right?
44
00:02:43 --> 00:02:50
And you have a hint
about these things.
45
00:02:50 --> 00:02:55
So we have a symmetric matrix
and the beauty is that it
46
00:02:55 --> 00:02:58
brings together all
of linear algebra.
47
00:02:58 --> 00:03:01
Including elimination,
that's when we see pivots.
48
00:03:01 --> 00:03:04
Including determinants
which are closely
49
00:03:04 --> 00:03:05
related to the pivots.
50
00:03:05 --> 00:03:08
And what do I mean
by upper left?
51
00:03:08 --> 00:03:13
I mean that if I have a three
by three symmetric matrix and I
52
00:03:13 --> 00:03:17
want to test it for positive
definite, and I guess actually
53
00:03:17 --> 00:03:22
this would be the easiest test
if I had a tiny matrix, three
54
00:03:22 --> 00:03:27
by three, and I had numbers
then this would be a good test.
55
00:03:27 --> 00:03:30
The determinants, by upper
left determinants I mean
56
00:03:30 --> 00:03:33
that one by one determinant.
57
00:03:33 --> 00:03:36
So just that first number
has to be positive.
58
00:03:36 --> 00:03:39
Then the two by two
determinant, that times
59
00:03:39 --> 00:03:42
that minus that times
that has to be positive.
60
00:03:42 --> 00:03:44
Oh I've already
been saying that.
61
00:03:44 --> 00:03:46
Let me just put
in some letters.
62
00:03:46 --> 00:03:48
So a has to be positive.
63
00:03:48 --> 00:03:51
This is symmetric, so
a times c has to be
64
00:03:51 --> 00:03:54
bigger than b squared.
65
00:03:54 --> 00:03:57
So that will tell us.
66
00:03:57 --> 00:03:59
And then for two
by two we finish.
67
00:03:59 --> 00:04:03
For three by three we would
also require the three by three
68
00:04:03 --> 00:04:05
determinant to be positive.
69
00:04:05 --> 00:04:08
But already here you're
seeing one point about a
70
00:04:08 --> 00:04:11
positive definite matrix.
71
00:04:11 --> 00:04:14
Its diagonal will
have to be positive.
72
00:04:14 --> 00:04:20
And somehow its diagonal has to
be not just above zero, but
73
00:04:20 --> 00:04:25
somehow it has to
defeat b squared.
74
00:04:25 --> 00:04:31
So the diagonal has to be
somehow more positive than
75
00:04:31 --> 00:04:35
whatever negative stuff might
come from off the diagonal.
76
00:04:35 --> 00:04:41
That's why I would
need a*c > b squared.
77
00:04:41 --> 00:04:43
So both of those will
be positive and their
78
00:04:43 --> 00:04:48
product has to be bigger
than the other guy.
79
00:04:48 --> 00:04:52
And then finally, a third
test is that all the
80
00:04:52 --> 00:04:53
eigenvalues are positive.
81
00:04:53 --> 00:04:56
And of course if I give you a
three by three matrix, that's
82
00:04:56 --> 00:04:59
probably not the easiest
test since you'd have to
83
00:04:59 --> 00:05:00
find the eigenvalues.
84
00:05:00 --> 00:05:05
Much easier to find the
determinants or the pivots.
85
00:05:05 --> 00:05:09
Actually, just while I'm
at it, so the first pivot
86
00:05:09 --> 00:05:12
of course is a itself.
87
00:05:12 --> 00:05:15
No difficulty there.
88
00:05:15 --> 00:05:18
The second pivot turns
out to be the ratio of
89
00:05:18 --> 00:05:22
a*c - b squared to a.
90
00:05:22 --> 00:05:25
So the connection between
pivots and determinants
91
00:05:25 --> 00:05:28
is just really close.
92
00:05:28 --> 00:05:30
Pivots are ratios
of determinants if
93
00:05:30 --> 00:05:31
you work it out.
94
00:05:31 --> 00:05:36
The second pivot, maybe I would
call that d_2, is the ratio
95
00:05:36 --> 00:05:40
of a*c - b squared over a.
96
00:05:40 --> 00:05:41
In other words it's
(c - b squared)/a.
97
00:05:41 --> 00:05:48
98
00:05:48 --> 00:05:51
Determinants are positive
and vice versa.
99
00:05:51 --> 00:05:54
Then it's fantastic that
the eigenvalues come
100
00:05:54 --> 00:05:56
into the picture.
101
00:05:56 --> 00:06:00
So those are three ways, three
important properties of a
102
00:06:00 --> 00:06:02
positive definite matrix.
103
00:06:02 --> 00:06:08
But I'd like to make the
definition something different.
104
00:06:08 --> 00:06:11
Now I'm coming to the meaning.
105
00:06:11 --> 00:06:14
If I think of those as
the tests, that's done.
106
00:06:14 --> 00:06:24
Now the meaning of
positive definite.
107
00:06:24 --> 00:06:27
The meaning of positive
definite and the applications
108
00:06:27 --> 00:06:32
are closely related to
looking for a minimum.
109
00:06:32 --> 00:06:39
And so what I'm going to bring
in here, so it's symmetric.
110
00:06:39 --> 00:06:47
Now for a symmetric matrix I
want to introduce the energy.
111
00:06:47 --> 00:06:51
So this is the reason why it
has so many applications and
112
00:06:51 --> 00:06:56
such important physical meaning
is that what I'm about to
113
00:06:56 --> 00:07:02
introduce, which is a function
of x, and here it is, it's x
114
00:07:02 --> 00:07:11
transpose times A, not A, I'm
sticking with K for
115
00:07:11 --> 00:07:16
my matrix, times x.
116
00:07:16 --> 00:07:20
I think of that as some f(x).
117
00:07:20 --> 00:07:24
And let's just see what it
would be if the matrix was
118
00:07:24 --> 00:07:29
two by two, [a, b; b, c].
119
00:07:29 --> 00:07:33
Suppose that's my matrix.
120
00:07:33 --> 00:07:36
We want to get a handle on
what, this is the first time
121
00:07:36 --> 00:07:42
I've ever written something
that has x's times x's.
122
00:07:42 --> 00:07:45
So it's going to be quadratic.
123
00:07:45 --> 00:07:48
They're going to
be x's times x's.
124
00:07:48 --> 00:07:52
And x is a general vector
of the right size so it's
125
00:07:52 --> 00:07:53
got components x_1, x_2.
126
00:07:55 --> 00:07:57
And there it's transpose,
so it's a row.
127
00:07:57 --> 00:08:01
And now I put in
the [a, b; b, c].
128
00:08:01 --> 00:08:05
And then I put in x again.
129
00:08:05 --> 00:08:08
This is going to give me
a very nice, simple,
130
00:08:08 --> 00:08:11
important expression.
131
00:08:11 --> 00:08:12
Depending on x_1 and x_2.
132
00:08:14 --> 00:08:18
Now what is, can we do
that multiplication?
133
00:08:18 --> 00:08:24
Maybe above I'll do the
multiplication of this
134
00:08:24 --> 00:08:28
pair and then I have the
other guy to bring in.
135
00:08:28 --> 00:08:30
So here, that would
be ax_1+bx_2.
136
00:08:30 --> 00:08:35
137
00:08:35 --> 00:08:36
And this would be bx_1+cx_2.
138
00:08:36 --> 00:08:39
139
00:08:39 --> 00:08:44
So that's the first,
that's this times this.
140
00:08:44 --> 00:08:45
What am I going to get?
141
00:08:45 --> 00:08:48
What shape, what size is
this result going to be?
142
00:08:48 --> 00:08:56
This K is n by n. x is a column
vector. n by one. x transpose,
143
00:08:56 --> 00:08:58
what's the shape
of x transpose?
144
00:08:58 --> 00:09:00
One by n?
145
00:09:00 --> 00:09:02
So what's the total result?
146
00:09:02 --> 00:09:03
One by one.
147
00:09:03 --> 00:09:04
Just a number.
148
00:09:04 --> 00:09:05
Just a function.
149
00:09:05 --> 00:09:06
It's a number.
150
00:09:06 --> 00:09:11
But it depends on the x's
and the matrix inside.
151
00:09:11 --> 00:09:12
Can we do it now?
152
00:09:12 --> 00:09:16
So I've got this to
multiply by this.
153
00:09:16 --> 00:09:19
Do you see an x_1
squared showing up?
154
00:09:19 --> 00:09:21
Yes, from there times there.
155
00:09:21 --> 00:09:24
And what's it multiplied by?
156
00:09:24 --> 00:09:26
The a.
157
00:09:26 --> 00:09:30
The first term is this times
the ax_1 is a(x_1 squared).
158
00:09:30 --> 00:09:33
So that's our first quadratic.
159
00:09:33 --> 00:09:35
Now there'd be an x_1, x_2.
160
00:09:36 --> 00:09:39
Let me leave that for a minute
and find the x_2 squared
161
00:09:39 --> 00:09:41
because it's easy.
162
00:09:41 --> 00:09:43
So where am I going
to get x_2 squared?
163
00:09:43 --> 00:09:47
I'm going to get that from
x_2, second guy here
164
00:09:47 --> 00:09:49
times second guy here.
165
00:09:49 --> 00:09:54
There's a c(x_2 squared).
166
00:09:54 --> 00:09:58
So you're seeing already
where the diagonal shows up.
167
00:09:58 --> 00:10:02
The diagonal a, c, whatever
is multiplying the
168
00:10:02 --> 00:10:04
perfect squares.
169
00:10:04 --> 00:10:08
And it'll be the off-diagonal
that multiplies the x_1, x_2.
170
00:10:08 --> 00:10:11
We might call those
the crossterms.
171
00:10:11 --> 00:10:13
And what do we get
for that then?
172
00:10:13 --> 00:10:16
We have x_1 times this guy.
173
00:10:16 --> 00:10:20
So that's a crossterm.
bx_1*x_2, right?
174
00:10:20 --> 00:10:24
And here's another one coming
from x_2 times this guy.
175
00:10:24 --> 00:10:27
And what's that one?
176
00:10:27 --> 00:10:29
It's also bx_1*x_2.
177
00:10:30 --> 00:10:34
So x_1, multiply that, x_2
multiply that, and so what
178
00:10:34 --> 00:10:37
do we have for this
crossterm here?
179
00:10:37 --> 00:10:38
Two of them.
180
00:10:39 --> 00:10:40
2bx_1*x_2.
181
00:10:43 --> 00:10:48
In other words, that b
and that b came together
182
00:10:48 --> 00:10:49
in the 2bx_1*x_2.
183
00:10:50 --> 00:10:56
So here's my energy.
184
00:10:56 --> 00:10:58
Can I just loosely
call it energy?
185
00:10:58 --> 00:11:02
And then as we get to
applications we'll see why.
186
00:11:02 --> 00:11:06
So I'm interested in
that because it has
187
00:11:06 --> 00:11:12
important meaning.
188
00:11:12 --> 00:11:15
Well, so now I'm ready
to define positive
189
00:11:15 --> 00:11:16
definite matrices.
190
00:11:16 --> 00:11:19
So I'll call that number four.
191
00:11:19 --> 00:11:23
But I'm going to
give it a big star.
192
00:11:23 --> 00:11:29
Even more because it's
the sort of key.
193
00:11:29 --> 00:11:34
So the test will be, you can
probably guess it, I look at
194
00:11:34 --> 00:11:40
this expression, that
x transpose Ax.
195
00:11:41 --> 00:11:46
And if it's a positive definite
matrix and this represents
196
00:11:46 --> 00:11:50
energy, the key will be that
this should be positive.
197
00:11:50 --> 00:11:57
This one should be
positive for all x's.
198
00:11:57 --> 00:11:59
Well, with one
exception, of course.
199
00:11:59 --> 00:12:06
All x's except, which vector
is it? x=0 would just
200
00:12:06 --> 00:12:10
give me-- See, I put K.
201
00:12:10 --> 00:12:17
My default for a matrix,
but should be, it's K.
202
00:12:17 --> 00:12:22
Except x=0, except
the zero vector.
203
00:12:22 --> 00:12:23
Of course.
204
00:12:23 --> 00:12:29
If x_1 and x_2 are both zero.
205
00:12:29 --> 00:12:34
Now that looks a little maybe
less straightforward, I would
206
00:12:34 --> 00:12:38
say, because it's a statement
about this is true
207
00:12:38 --> 00:12:40
for all x_1 and x_2.
208
00:12:41 --> 00:12:44
And we better do some
examples and draw a picture.
209
00:12:44 --> 00:12:51
Let me draw a
picture right away.
210
00:12:51 --> 00:12:54
So here's x_1 direction.
211
00:12:54 --> 00:12:56
Here's x_2 direction.
212
00:12:56 --> 00:13:05
And here is the x transpose
Ax, my function.
213
00:13:05 --> 00:13:09
So this depends on
two variables.
214
00:13:09 --> 00:13:13
So it's going to be a sort
of a surface if I draw it.
215
00:13:13 --> 00:13:16
Now, what point do
we absolutely know?
216
00:13:16 --> 00:13:21
And I put A again.
217
00:13:21 --> 00:13:29
I am so sorry.
218
00:13:29 --> 00:13:30
Well, we know one point.
219
00:13:30 --> 00:13:34
It's there whatever
that matrix might be.
220
00:13:34 --> 00:13:35
It's there.
221
00:13:35 --> 00:13:37
Zero, right?
222
00:13:37 --> 00:13:40
You just told me that if
both x's are zero then we
223
00:13:40 --> 00:13:42
automatically get zero.
224
00:13:42 --> 00:13:47
Now what do you think the shape
of this curve, the shape of
225
00:13:47 --> 00:13:51
this graph is going
to look like?
226
00:13:51 --> 00:13:54
The point is, if we're
positive definite now.
227
00:13:54 --> 00:13:58
So I'm drawing the picture
for positive definite.
228
00:13:58 --> 00:14:04
So my definition is that
the energy goes up.
229
00:14:04 --> 00:14:06
It's positive, right?
230
00:14:06 --> 00:14:10
When I leave, when I move away
from that point I go upwards.
231
00:14:10 --> 00:14:13
That point will be a minimum.
232
00:14:13 --> 00:14:17
Let me just draw it roughly.
233
00:14:17 --> 00:14:23
So it sort of goes
up like this.
234
00:14:23 --> 00:14:30
These cheap 2-D boards and
I've got a three-dimensional
235
00:14:30 --> 00:14:34
picture here.
236
00:14:34 --> 00:14:35
But you see it somehow?
237
00:14:35 --> 00:14:40
What word or what's
your visualization?
238
00:14:40 --> 00:14:43
It has a minimum there.
239
00:14:43 --> 00:14:46
That's why minimization, which
was like, the core problem
240
00:14:46 --> 00:14:49
in calculus, is here now.
241
00:14:49 --> 00:14:54
But for functions of
two x's or n x's.
242
00:14:54 --> 00:14:58
We're up the dimension
over the basic minimum
243
00:14:58 --> 00:15:03
problem of calculus.
244
00:15:03 --> 00:15:06
It's sort of like a parabola
It's cross-sections cutting
245
00:15:06 --> 00:15:09
down through the thing would
be just parabolas because
246
00:15:09 --> 00:15:10
of the x squared.
247
00:15:10 --> 00:15:12
I'm going to call this a bowl.
248
00:15:12 --> 00:15:16
It's a short word.
249
00:15:16 --> 00:15:16
Do you see it?
250
00:15:16 --> 00:15:18
It opens up.
251
00:15:18 --> 00:15:20
That's the key point,
that it opens upward.
252
00:15:20 --> 00:15:22
And let's do some examples.
253
00:15:22 --> 00:15:26
Tell me some positive definite.
254
00:15:26 --> 00:15:30
So positive definite and then
let me here put some not
255
00:15:30 --> 00:15:35
positive definite cases.
256
00:15:35 --> 00:15:38
Tell me a matrix.
257
00:15:38 --> 00:15:42
Well, what's the easiest, first
matrix that occurs to you as
258
00:15:42 --> 00:15:44
a positive definite matrix?
259
00:15:44 --> 00:15:49
The identity.
260
00:15:49 --> 00:15:52
That passes all our tests,
its eigenvalues are one,
261
00:15:52 --> 00:15:54
its pivots are one, the
determinants are one.
262
00:15:54 --> 00:16:00
And the function is x_1
squared plus x_2 squared
263
00:16:00 --> 00:16:05
with no b in it.
264
00:16:05 --> 00:16:08
It's just a perfect bowl,
perfectly symmetric, the
265
00:16:08 --> 00:16:12
way it would come off
a potter's wheel.
266
00:16:12 --> 00:16:16
Now let me take one that's
maybe not so, let me
267
00:16:16 --> 00:16:18
put a nine there.
268
00:16:18 --> 00:16:20
So I'm off to a
reasonable start.
269
00:16:20 --> 00:16:22
I have an x_1 squared
and a nine x_2 squared.
270
00:16:24 --> 00:16:27
And now I want to ask you, what
could I put in there that would
271
00:16:27 --> 00:16:30
leave it positive definite?
272
00:16:30 --> 00:16:33
Well, give me a couple
of possibilities.
273
00:16:33 --> 00:16:37
What's a nice, not too big
now, that's the thing.
274
00:16:37 --> 00:16:38
Two.
275
00:16:38 --> 00:16:39
Two would be fine.
276
00:16:39 --> 00:16:42
So if I had a two there and a
two there I would have a
277
00:16:42 --> 00:16:47
4x_1*x_2 and it would, like,
this, instead of being a
278
00:16:47 --> 00:16:55
circle, which it was for the
identity, the plane there would
279
00:16:55 --> 00:16:59
cut out a ellipse instead.
280
00:16:59 --> 00:17:02
But it would be a good ellipse.
281
00:17:02 --> 00:17:06
Because we're doing squares,
we're really, the Greeks
282
00:17:06 --> 00:17:11
understood these second degree
things and they would have
283
00:17:11 --> 00:17:18
known this would have
been an ellipse.
284
00:17:18 --> 00:17:23
How high can I go with that two
or where do I have to stop?
285
00:17:23 --> 00:17:27
Where would I have to, if I
wanted to change the two, let
286
00:17:27 --> 00:17:33
me just focus on that one,
suppose I wanted to change it.
287
00:17:33 --> 00:17:36
First of all, give me
one that's, how about
288
00:17:36 --> 00:17:38
the borderline.
289
00:17:38 --> 00:17:40
Three would be the borderline.
290
00:17:40 --> 00:17:41
Why's that?
291
00:17:41 --> 00:17:48
Because at three we have nine
minus nine for the determinant.
292
00:17:48 --> 00:17:51
So the determinant is zero.
293
00:17:51 --> 00:17:53
Of course it passed
the first test.
294
00:17:53 --> 00:17:54
One by one was ok.
295
00:17:54 --> 00:18:03
But two by two was not,
was at the borderline.
296
00:18:03 --> 00:18:07
What else should I think?
297
00:18:07 --> 00:18:11
Oh, that's a very
interesting case.
298
00:18:11 --> 00:18:13
The borderline.
299
00:18:13 --> 00:18:15
You know, it almost makes it.
300
00:18:15 --> 00:18:21
But can you tell me the
eigenvalues of that matrix?
301
00:18:21 --> 00:18:25
Don't do any
quadratic equations.
302
00:18:25 --> 00:18:28
How do I know, what's one
eigenvalue of a matrix?
303
00:18:28 --> 00:18:30
You made it singular, right?
304
00:18:30 --> 00:18:31
You made that matrix singular.
305
00:18:31 --> 00:18:32
Determinant zero.
306
00:18:32 --> 00:18:36
So one of its
eigenvalues is zero.
307
00:18:36 --> 00:18:40
And the other one is visible
by looking at the trace.
308
00:18:40 --> 00:18:44
I just quickly mentioned that
if I add the diagonal, I
309
00:18:44 --> 00:18:47
get the same answer as if I
add the two eigenvalues.
310
00:18:47 --> 00:18:51
So that other eigenvalue
must be ten.
311
00:18:51 --> 00:18:55
And this is entirely typical,
that ten and zero, the extreme
312
00:18:55 --> 00:19:01
eigenvalues, lambda_max and
lambda_min, are bigger than,
313
00:19:01 --> 00:19:04
these diagonal guys are inside.
314
00:19:04 --> 00:19:09
They're inside, between zero
and ten and it's these terms
315
00:19:09 --> 00:19:13
that enter somehow and gave
us an eigenvalue of ten
316
00:19:13 --> 00:19:15
and an eigenvalue of zero.
317
00:19:15 --> 00:19:20
I guess I'm tempted to
try to draw that figure.
318
00:19:20 --> 00:19:25
Just to get a feeling of
what's with that one.
319
00:19:25 --> 00:19:30
It always helps to get
the borderline case.
320
00:19:30 --> 00:19:32
So what's with this one?
321
00:19:32 --> 00:19:35
Let me see what my
quadratic would be.
322
00:19:35 --> 00:19:37
Can I just change it up here?
323
00:19:37 --> 00:19:38
Rather than rewriting it.
324
00:19:38 --> 00:19:42
So I'm going to, I'll
put it up here.
325
00:19:42 --> 00:19:46
So I have to change
that four to what?
326
00:19:46 --> 00:19:48
Now that I'm looking
at this matrix.
327
00:19:48 --> 00:19:51
That four is now a six.
328
00:19:51 --> 00:19:53
Six.
329
00:19:53 --> 00:19:56
This is my guy for this one.
330
00:19:56 --> 00:19:58
Which is not positive definite.
331
00:19:58 --> 00:20:00
Let me tell you right away
the word that I would
332
00:20:00 --> 00:20:02
use for this one.
333
00:20:02 --> 00:20:06
I would call it positive
semi-definite because it's
334
00:20:06 --> 00:20:09
almost there, but not quite.
335
00:20:09 --> 00:20:15
So semi-definite allows the
matrix to be singular.
336
00:20:15 --> 00:20:19
So semi-definite, maybe
I'll do it in green what
337
00:20:19 --> 00:20:22
semi-definite would be.
338
00:20:22 --> 00:20:31
Semi-def would be eigenvalues
greater than or equal zero.
339
00:20:31 --> 00:20:35
Determinants greater
than or equal zero.
340
00:20:35 --> 00:20:39
Pivots greater than zero
if they're there or then
341
00:20:39 --> 00:20:41
we run out of pivots.
342
00:20:41 --> 00:20:44
You could say greater than
or equal to zero then.
343
00:20:44 --> 00:20:48
And energy, greater
than or equal to zero
344
00:20:48 --> 00:20:53
for semi-definite.
345
00:20:53 --> 00:20:58
And when would the energy, what
x's, what would be the like,
346
00:20:58 --> 00:21:02
you could say the ground states
or something, what x's, so
347
00:21:02 --> 00:21:06
greater than or equal to zero,
emphasize that possibility
348
00:21:06 --> 00:21:10
of equal in the
semi-definite case.
349
00:21:10 --> 00:21:17
Suppose I have a semi-definite
matrix, yeah, I've got one.
350
00:21:17 --> 00:21:19
But it's singular.
351
00:21:19 --> 00:21:26
So that means a singular matrix
takes some vector x to zero.
352
00:21:26 --> 00:21:27
Right?
353
00:21:27 --> 00:21:30
If my matrix is actually
singular, then there'll be
354
00:21:30 --> 00:21:32
an x where Kx is zero.
355
00:21:32 --> 00:21:35
And then, of course,
multiplying by x transpose,
356
00:21:35 --> 00:21:36
I'm still at zero.
357
00:21:36 --> 00:21:41
So the x's, the zero energy
guys, this is straightforward,
358
00:21:41 --> 00:21:49
the zero energy guys, the ones
where x transpose Kx is zero,
359
00:21:49 --> 00:21:56
will happen when Kx is zero.
360
00:21:56 --> 00:22:03
If Kx is zero, and we'll
see it in that example.
361
00:22:03 --> 00:22:05
Let's see it in that example.
362
00:22:05 --> 00:22:12
What's the x for which,
I could say in the null
363
00:22:12 --> 00:22:18
space, what's the vector
x that that matrix kills?
364
00:22:18 --> 00:22:21
365
00:22:21 --> 00:22:23
, right?
366
00:22:23 --> 00:22:24
The vector .
367
00:22:24 --> 00:22:28
368
00:22:28 --> 00:22:30
gives me .
369
00:22:30 --> 00:22:33
That's the vector that,
so I get 3-3, the
370
00:22:33 --> 00:22:36
zero, 9-9, the zero.
371
00:22:36 --> 00:22:42
So I believe that this
thing will be-- Is it
372
00:22:42 --> 00:22:44
zero at three, minus one?
373
00:22:44 --> 00:22:47
I think that it
has to be, right?
374
00:22:47 --> 00:22:52
If I take x_1 to be three and
x_2 to be minus one, I think
375
00:22:52 --> 00:22:54
I've got zero energy here.
376
00:22:54 --> 00:22:58
Do I? x_1 squared will be
at the nine and nine x_2
377
00:22:58 --> 00:23:01
squared will be nine more.
378
00:23:01 --> 00:23:02
And what will be this 6x_1*x_2?
379
00:23:04 --> 00:23:09
What will that come out
for this x_1 and x_2?
380
00:23:09 --> 00:23:10
Minus 18.
381
00:23:10 --> 00:23:11
Had to, right?
382
00:23:11 --> 00:23:13
So I'd get nine from
there, nine from
383
00:23:13 --> 00:23:15
there, minus 18, zero.
384
00:23:15 --> 00:23:18
So the graph for this
positive semi-definite
385
00:23:18 --> 00:23:21
will look a bit like this.
386
00:23:21 --> 00:23:26
There'll be a direction in
which it doesn't climb.
387
00:23:26 --> 00:23:29
It doesn't go below
the base, right?
388
00:23:29 --> 00:23:31
It's never negative.
389
00:23:31 --> 00:23:33
This is now the
semi-definite picture.
390
00:23:33 --> 00:23:36
But it can run along the base.
391
00:23:36 --> 00:23:40
And it will for the vector
x_1=3, x_2=-1, I don't know
392
00:23:40 --> 00:23:45
where that is, one, two, three,
and then maybe minus one.
393
00:23:45 --> 00:23:51
Along some line here the
graph doesn't go up.
394
00:23:51 --> 00:23:56
It's sitting, can you imagine
that sitting in the base?
395
00:23:56 --> 00:24:05
I'm not Rembrandt here, but in
the other direction it goes up.
396
00:24:05 --> 00:24:08
Oh, the hell with that one.
397
00:24:08 --> 00:24:10
Do you see, sort of?
398
00:24:10 --> 00:24:14
It's like a trough,
would you say?
399
00:24:14 --> 00:24:16
I mean, it's like a,
you know, a bit of a
400
00:24:16 --> 00:24:19
drainpipe or something.
401
00:24:19 --> 00:24:27
It's running along the ground,
along this direction and
402
00:24:27 --> 00:24:30
in the other directions
it does go up.
403
00:24:30 --> 00:24:35
So it's shaped like this
with the base not climbing.
404
00:24:35 --> 00:24:39
Whereas here, there's
no bad direction.
405
00:24:39 --> 00:24:40
Climbs every way you go.
406
00:24:40 --> 00:24:45
So that's positive definite and
that's positive semi-definite.
407
00:24:45 --> 00:24:50
Well suppose I push
it a little further.
408
00:24:50 --> 00:24:56
Let me make a place here for
a matrix that isn't even
409
00:24:56 --> 00:25:01
positive semi-definite.
410
00:25:01 --> 00:25:05
Now it's just going to
go down somewhere.
411
00:25:05 --> 00:25:07
I'll start again with one
and nine and tell me
412
00:25:07 --> 00:25:09
what to put in now.
413
00:25:09 --> 00:25:13
So this is going to be a
case where the off-diagonal
414
00:25:13 --> 00:25:15
is too big, it wins.
415
00:25:15 --> 00:25:18
And prevents positive definite.
416
00:25:18 --> 00:25:21
So what number would
you like here?
417
00:25:21 --> 00:25:22
Five?
418
00:25:22 --> 00:25:27
Five is certainly plenty.
419
00:25:27 --> 00:25:30
So now I have [1, 5; 5, 9].
420
00:25:30 --> 00:25:37
Let me take a little space on
a board just to show you.
421
00:25:37 --> 00:25:42
Sorry about that.
422
00:25:42 --> 00:25:46
So I'm going to do the [1, 5;
5, 9] just because they're all
423
00:25:46 --> 00:25:48
important, but then
we're coming back to
424
00:25:48 --> 00:25:49
positive definite.
425
00:25:49 --> 00:25:59
So if it's [1, 5; 5, 9] and I
do that usual x, x transpose Kx
426
00:25:59 --> 00:26:03
and I do the multiplication
out, I see the one x_1 squared
427
00:26:03 --> 00:26:06
and I see the nine
x_2 squareds.
428
00:26:06 --> 00:26:11
And how many
x_1*x_2's do I see?
429
00:26:11 --> 00:26:15
Five from there, five
from there, ten.
430
00:26:15 --> 00:26:20
And I believe that
can be negative.
431
00:26:20 --> 00:26:24
The fact of having all nice
plus signs is not going to help
432
00:26:24 --> 00:26:28
it because we can choose, as we
already did, x_1 to be like
433
00:26:28 --> 00:26:31
a negative number and
x_2 to be a positive.
434
00:26:31 --> 00:26:35
And we can get this guy to be
negative and make it, in this
435
00:26:35 --> 00:26:41
case we can make it defeat
these positive parts.
436
00:26:41 --> 00:26:43
What choice would do it?
437
00:26:43 --> 00:26:49
Let me take x_1 to be minus one
and tell me an x_2 that's good
438
00:26:49 --> 00:26:55
enough to show that this thing
is not positive definite or
439
00:26:55 --> 00:26:58
even semi-definite,
it goes downhill.
440
00:26:58 --> 00:26:59
Take x_2 equal?
441
00:26:59 --> 00:27:04
What do you say?
442
00:27:04 --> 00:27:05
1/2?
443
00:27:05 --> 00:27:08
Yeah, I don't want too big an
x_2 because if I have too big
444
00:27:08 --> 00:27:10
an x_2, then this'll
be important.
445
00:27:10 --> 00:27:14
Does 1/2 do it?
446
00:27:14 --> 00:27:18
So I've got 1/4, that's
positive, but not very.
447
00:27:18 --> 00:27:23
9/4, so I'm up to 10/4,
but this guy is what?
448
00:27:23 --> 00:27:26
Ten and the minus
is minus five.
449
00:27:26 --> 00:27:27
Yeah.
450
00:27:27 --> 00:27:31
So that absolutely goes,
at this one I come
451
00:27:31 --> 00:27:35
out less than zero.
452
00:27:35 --> 00:27:37
And I might as well complete.
453
00:27:37 --> 00:27:43
So this is the case where I
would call it indefinite.
454
00:27:43 --> 00:27:45
Indefinite.
455
00:27:45 --> 00:27:50
It goes up like if x_2 is
zero, then it's just got
456
00:27:50 --> 00:27:52
x_1 squared, that's up.
457
00:27:52 --> 00:27:55
If x_1 is zero, it's only
got x_2 squared, that's up.
458
00:27:55 --> 00:27:58
But there are other directions
where it goes downhill.
459
00:27:58 --> 00:28:01
So it goes either up, it
goes both up in some
460
00:28:01 --> 00:28:03
ways and down in others.
461
00:28:03 --> 00:28:08
And what kind of a graph, what
kind of a surface would I now
462
00:28:08 --> 00:28:15
have for x transpose for this x
transpose, this indefinite guy?
463
00:28:15 --> 00:28:26
So up in some ways
and down in others.
464
00:28:26 --> 00:28:35
This gets really hard to draw.
465
00:28:35 --> 00:28:40
I believe that if you ride
horses you have an edge
466
00:28:40 --> 00:28:43
on visualizing this.
467
00:28:43 --> 00:28:45
So it's called, what kind
of a point's it called?
468
00:28:45 --> 00:28:50
Saddle point, it's
called a saddle point.
469
00:28:50 --> 00:28:53
So what's a saddle point?
470
00:28:53 --> 00:28:56
That's not bad, right?
471
00:28:56 --> 00:28:58
So this is a direction
where it went up.
472
00:28:58 --> 00:29:02
This is a direction
where it went down.
473
00:29:02 --> 00:29:09
And so it sort of
fills in somehow.
474
00:29:09 --> 00:29:18
Or maybe, if you don't, I
mean, who rides horses now?
475
00:29:18 --> 00:29:25
Actually maybe something we do
do is drive over mountains.
476
00:29:25 --> 00:29:35
So the path, if the road is
sort of well-chosen, the road
477
00:29:35 --> 00:29:42
will go, it'll look for the,
this would be-- Yeah,
478
00:29:42 --> 00:29:43
here's our road.
479
00:29:43 --> 00:29:46
We would do as little
climbing as possible.
480
00:29:46 --> 00:29:48
The mountain would go
like this, sort of.
481
00:29:48 --> 00:29:53
So this would be like, the
bottom part looking along
482
00:29:53 --> 00:29:55
the peaks of the mountains.
483
00:29:55 --> 00:29:59
But it's the top part looking
along the driving direction.
484
00:29:59 --> 00:30:05
So driving, it's a maximum,
but in the mountain range
485
00:30:05 --> 00:30:06
direction it's a minimum.
486
00:30:06 --> 00:30:10
So it's a saddle point.
487
00:30:10 --> 00:30:14
So that's what you get from
a typical symmetric matrix.
488
00:30:14 --> 00:30:19
And if it was minus five it
would still be the same saddle
489
00:30:19 --> 00:30:25
point, would still be 9-25, it
would still be negative
490
00:30:25 --> 00:30:27
and a saddle.
491
00:30:27 --> 00:30:30
Positive guys are our thing.
492
00:30:30 --> 00:30:32
Alright.
493
00:30:32 --> 00:30:36
So now back to
positive definite.
494
00:30:36 --> 00:30:40
With these four tests
and then the discussion
495
00:30:40 --> 00:30:45
of semi-definite.
496
00:30:45 --> 00:30:49
Very key, that energy.
497
00:30:49 --> 00:30:51
Let me just look
ahead a moment.
498
00:30:51 --> 00:30:58
Most physical problems, many,
many physical problems,
499
00:30:58 --> 00:31:00
you have an option.
500
00:31:00 --> 00:31:04
Either you solve some
equations, either you find
501
00:31:04 --> 00:31:10
the solution from our
equations, Ku=f, typically.
502
00:31:10 --> 00:31:12
Matrix equation or
differential equation.
503
00:31:12 --> 00:31:20
Or there's another option of
minimizing some function.
504
00:31:20 --> 00:31:23
Some energy.
505
00:31:23 --> 00:31:25
And it gives the
same equations.
506
00:31:25 --> 00:31:32
So this minimizing energy
will be a second way to
507
00:31:32 --> 00:31:36
describe the applications.
508
00:31:36 --> 00:31:40
Now can I get a number five?
509
00:31:40 --> 00:31:44
There's an important number
five and then you know
510
00:31:44 --> 00:31:48
really all you need to know
about symmetric matrices.
511
00:31:48 --> 00:31:51
This gives me, about positive
definite matrices, this
512
00:31:51 --> 00:32:01
gives me a chance to recap.
513
00:32:01 --> 00:32:06
So I'm going to put
down a number five.
514
00:32:06 --> 00:32:16
Because this is where
the matrices come from.
515
00:32:16 --> 00:32:17
Really important.
516
00:32:17 --> 00:32:20
And it's where they'll come
from in all these applications
517
00:32:20 --> 00:32:23
that chapter two is going to be
all about, that we're
518
00:32:23 --> 00:32:25
going to start.
519
00:32:25 --> 00:32:29
So they come, these positive
definite matrices, so this is
520
00:32:29 --> 00:32:36
another way to, it's a test for
positive definite matrices
521
00:32:36 --> 00:32:40
and it's, actually, it's
where they come from.
522
00:32:40 --> 00:32:44
So here's a positive
definite matrix.
523
00:32:44 --> 00:32:52
They come from A transpose A.
524
00:32:52 --> 00:32:56
A fundamental message is that
if I have just an average
525
00:32:56 --> 00:33:00
matrix, possibly rectangular,
could be a square but not
526
00:33:00 --> 00:33:07
symmetric, then sooner or
later, in fact usually sooner,
527
00:33:07 --> 00:33:10
you end up looking
at A transpose A.
528
00:33:10 --> 00:33:11
We've seen that already.
529
00:33:11 --> 00:33:15
And we already know that A
transpose A is square, we
530
00:33:15 --> 00:33:18
already know it's symmetric and
now we're going to know that
531
00:33:18 --> 00:33:20
it's positive definite.
532
00:33:20 --> 00:33:25
So matrices like A transpose
A are positive definite or
533
00:33:25 --> 00:33:28
possibly semi-definite.
534
00:33:28 --> 00:33:29
There's that possibility.
535
00:33:29 --> 00:33:32
If A was the zero matrix, of
course, we would just get the
536
00:33:32 --> 00:33:37
zero matrix which would be only
semi-definite, or other ways
537
00:33:37 --> 00:33:42
to get a semi-definite.
538
00:33:42 --> 00:33:46
So I'm saying that if K, if I
have a matrix, any matrix, and
539
00:33:46 --> 00:33:51
I form A transpose A, I get a
positive definite matrix or
540
00:33:51 --> 00:33:56
maybe just semi-definite,
but not indefinite.
541
00:33:56 --> 00:34:01
Can we see why?
542
00:34:01 --> 00:34:11
Why is this positive
definite or semi-?
543
00:34:11 --> 00:34:13
So that's my question.
544
00:34:13 --> 00:34:17
And the answer is really
worth, it's just neat
545
00:34:17 --> 00:34:19
and worth seeing.
546
00:34:19 --> 00:34:23
So do I want to look at the
pivots of A transpose A?
547
00:34:23 --> 00:34:25
No.
548
00:34:25 --> 00:34:29
They're something, but whatever
they are, I can't really
549
00:34:29 --> 00:34:30
follow those well.
550
00:34:30 --> 00:34:34
Or the eigenvalues very
well, or the determinants.
551
00:34:34 --> 00:34:36
None of those come out nicely.
552
00:34:36 --> 00:34:41
But the real guy
works perfectly.
553
00:34:41 --> 00:34:46
So look at x transpose Kx.
554
00:34:46 --> 00:34:48
555
00:34:48 --> 00:34:57
So I'm just doing, following
my instinct here.
556
00:34:57 --> 00:35:03
So if K is A transpose A, my
claim is, what am I saying
557
00:35:03 --> 00:35:07
then about this energy?
558
00:35:07 --> 00:35:13
What is it that I want to
discover and understand?
559
00:35:13 --> 00:35:15
Why it's positive.
560
00:35:15 --> 00:35:20
Why does taking any matrix,
multiplying by its transpose
561
00:35:20 --> 00:35:27
produce something
that's positive?
562
00:35:27 --> 00:35:31
Can you see any reason why that
quantity, which looks kind of
563
00:35:31 --> 00:35:38
messy, I just want to look at
it the right way to see why
564
00:35:38 --> 00:35:41
that should be positive, that
should come out positive.
565
00:35:41 --> 00:35:45
So I'm not going to get into
numbers, I'm not going to get
566
00:35:45 --> 00:35:47
into diagonals and
off-diagonals.
567
00:35:47 --> 00:35:53
I'm just going to do one thing
to understand that particular
568
00:35:53 --> 00:35:56
combination, x transpose
A transpose Ax.
569
00:35:57 --> 00:35:59
What shall I do?
570
00:35:59 --> 00:36:06
Anybody see what I might do?
571
00:36:06 --> 00:36:10
Yeah, you're seeing here
if you look at it again,
572
00:36:10 --> 00:36:12
what are you seeing here?
573
00:36:12 --> 00:36:14
Tell me again.
574
00:36:14 --> 00:36:20
If I take Ax together, then
what's the other half?
575
00:36:20 --> 00:36:23
It's the transpose of Ax.
576
00:36:23 --> 00:36:26
So I just want to write that
as, I just want to think
577
00:36:26 --> 00:36:27
of it that way, as Ax.
578
00:36:29 --> 00:36:31
And here's the transpose of Ax.
579
00:36:32 --> 00:36:33
Right?
580
00:36:33 --> 00:36:36
Because transposes of Ax, so
transpose guys in the opposite
581
00:36:36 --> 00:36:39
order, and the multiplication--
582
00:36:39 --> 00:36:41
This is the great.
583
00:36:41 --> 00:36:44
I call these proof by
parenthesis because I'm just
584
00:36:44 --> 00:36:51
putting parentheses in the
right place, but the key law
585
00:36:51 --> 00:36:57
of matrix multiplication is
that, that I can put (AB)C
586
00:36:57 --> 00:36:58
is the same as A(BC).
587
00:36:58 --> 00:37:01
588
00:37:01 --> 00:37:04
That rule, which is just
multiply it out and you see
589
00:37:04 --> 00:37:07
that parentheses are not needed
because if you keep them in the
590
00:37:07 --> 00:37:10
right order you can do this
first, or you can
591
00:37:10 --> 00:37:12
do this first.
592
00:37:12 --> 00:37:13
Same answer.
593
00:37:13 --> 00:37:15
What do I learn from that?
594
00:37:15 --> 00:37:17
What was the point?
595
00:37:17 --> 00:37:19
This is some vector, I don't
know especially what it
596
00:37:19 --> 00:37:21
is times its transpose.
597
00:37:21 --> 00:37:24
So that's the length squared.
598
00:37:24 --> 00:37:27
What's the key fact about that?
599
00:37:27 --> 00:37:30
That it is never negative.
600
00:37:30 --> 00:37:41
It's always greater than
zero or possibly equal.
601
00:37:41 --> 00:37:44
When does that
quantity equal zero?
602
00:37:44 --> 00:37:45
When Ax is zero.
603
00:37:45 --> 00:37:47
When Ax is zero.
604
00:37:47 --> 00:37:49
Because this is a vector.
605
00:37:49 --> 00:37:50
That's the same
vector transposed.
606
00:37:50 --> 00:37:52
And everybody's
got that picture.
607
00:37:52 --> 00:37:58
When I take any y transpose y,
I get y_1 squared plus y_2
608
00:37:58 --> 00:38:00
squared through y_n squared.
609
00:38:00 --> 00:38:05
And I get a positive answer
except if the vector is zero.
610
00:38:05 --> 00:38:11
So it's zero when Ax is zero.
611
00:38:11 --> 00:38:13
So that's going to be the key.
612
00:38:13 --> 00:38:19
If I pick any matrix A, and I
can just take an example, but
613
00:38:19 --> 00:38:22
chapter, the applications
are just going to be
614
00:38:22 --> 00:38:23
full of examples.
615
00:38:23 --> 00:38:29
Where the problem begins with a
matrix A and then A transpose
616
00:38:29 --> 00:38:34
shows up and it's the
combination A transpose
617
00:38:34 --> 00:38:36
A that we work with.
618
00:38:36 --> 00:38:40
And we're just learning that
it's positive definite.
619
00:38:40 --> 00:38:48
Unless, shall I just hang on
since I've got here, I have to
620
00:38:48 --> 00:38:53
say when is it, have to get
these two possibilities.
621
00:38:53 --> 00:38:56
Positive definite or
only semi-definite.
622
00:38:56 --> 00:39:05
So what's the key to that
borderline question?
623
00:39:05 --> 00:39:11
This thing will be only
semi-definite if there's
624
00:39:11 --> 00:39:12
a solution to Ax=0.
625
00:39:12 --> 00:39:16
626
00:39:16 --> 00:39:23
If there is an x, well, there's
always the zero vector.
627
00:39:23 --> 00:39:26
Zero vector I can't
expect to be positive.
628
00:39:26 --> 00:39:35
So I'm looking for if there's
an x so that Ax is zero but x
629
00:39:35 --> 00:39:48
is not zero, then I'll
only be semi-definite.
630
00:39:48 --> 00:39:50
That's the test.
631
00:39:50 --> 00:39:52
If there is a solution to Ax=0.
632
00:39:52 --> 00:39:55
633
00:39:55 --> 00:39:59
When we see applications
that'll mean there's a
634
00:39:59 --> 00:40:03
displacement with
no stretching.
635
00:40:03 --> 00:40:10
We might have a line of springs
and when could the line
636
00:40:10 --> 00:40:16
of springs displace
with no stretching?
637
00:40:16 --> 00:40:18
When it's free-free, right?
638
00:40:18 --> 00:40:24
If I have a line of springs and
no supports at the ends, then
639
00:40:24 --> 00:40:27
that would be the case where
it could shift over by
640
00:40:27 --> 00:40:29
the vector.
641
00:40:29 --> 00:40:33
So that would be the case where
the matrix is only singular.
642
00:40:33 --> 00:40:34
We know that.
643
00:40:34 --> 00:40:37
The matrix is now
positive semi-definite.
644
00:40:37 --> 00:40:38
We just learned that.
645
00:40:38 --> 00:40:46
So the free-free matrix, like
B, both ends free, or C.
646
00:40:46 --> 00:40:52
So our answer is going
to be that K and T are
647
00:40:52 --> 00:40:56
positive definite.
648
00:40:56 --> 00:40:59
And our other two guys, the
singular ones, of course,
649
00:40:59 --> 00:41:00
just don't make it.
650
00:41:00 --> 00:41:04
B at both ends, the free-free
line of springs, it can
651
00:41:04 --> 00:41:07
shift without stretching.
652
00:41:07 --> 00:41:11
Since Ax will measure the
stretching when it just shifts
653
00:41:11 --> 00:41:14
rigid motion, the Ax is
zero and we see only
654
00:41:14 --> 00:41:16
positive definite.
655
00:41:16 --> 00:41:19
And also C, the circular one.
656
00:41:19 --> 00:41:22
There it can displace with no
stretching because it can
657
00:41:22 --> 00:41:24
just turn in the circle.
658
00:41:24 --> 00:41:45
So these guys will be only
positive semi-definite.
659
00:41:45 --> 00:41:49
Maybe I better say
this another way.
660
00:41:49 --> 00:41:51
When is this positive definite?
661
00:41:51 --> 00:41:55
Can I use just a different
sentence to describe
662
00:41:55 --> 00:41:57
this possibility?
663
00:41:57 --> 00:42:05
This is positive definite
provided, so what I'm going to
664
00:42:05 --> 00:42:08
write now is to remove this
possibility and get
665
00:42:08 --> 00:42:10
positive definite.
666
00:42:10 --> 00:42:16
This is positive definite
provided, now, I could
667
00:42:16 --> 00:42:17
say it this way.
668
00:42:17 --> 00:42:25
The A has independent columns.
669
00:42:25 --> 00:42:28
So I just needed to give
you another way of looking
670
00:42:28 --> 00:42:33
at this Ax=0 question.
671
00:42:33 --> 00:42:37
If A has independent columns,
what does that mean?
672
00:42:37 --> 00:42:40
That means that the only
solution to Ax=0 is
673
00:42:40 --> 00:42:42
the zero solution.
674
00:42:42 --> 00:42:47
In other words, it means that
this thing works perfectly
675
00:42:47 --> 00:42:50
and gives me positive.
676
00:42:50 --> 00:42:53
When A has independent columns.
677
00:42:53 --> 00:42:56
Let's just remember
our K, T, B, C.
678
00:42:56 --> 00:43:08
So here's a matrix, so let me
take the T matrix, that's
679
00:43:08 --> 00:43:11
this one, this guy.
680
00:43:11 --> 00:43:15
And then the third
column is .
681
00:43:15 --> 00:43:19
Those three columns
are independent.
682
00:43:19 --> 00:43:21
They point off.
683
00:43:21 --> 00:43:23
They don't lie in a plane.
684
00:43:23 --> 00:43:27
They point off in three
different directions.
685
00:43:27 --> 00:43:34
And then there are no solutions
to, no x's that's go Kx=0.
686
00:43:34 --> 00:43:39
687
00:43:39 --> 00:43:41
So that would be a case
of independent columns.
688
00:43:41 --> 00:43:45
Let me make a case of
dependent columns.
689
00:43:45 --> 00:43:47
So and I'm going
to make it B now.
690
00:43:47 --> 00:43:51
Now the columns of that
matrix are dependent.
691
00:43:51 --> 00:43:54
There's a combination of
them that give zero.
692
00:43:54 --> 00:43:56
They all lie in the same plane.
693
00:43:56 --> 00:44:00
There's a solution to that
matrix times x equal zero.
694
00:44:00 --> 00:44:03
What combination of those
columns shows me that
695
00:44:03 --> 00:44:05
they are dependent?
696
00:44:05 --> 00:44:09
That some combination of those
three columns, some amount of
697
00:44:09 --> 00:44:12
this plus some amount of this
plus some amount of that column
698
00:44:12 --> 00:44:15
gives me the zero vector.
699
00:44:15 --> 00:44:17
You see the combination.
700
00:44:17 --> 00:44:21
What should I take?
again.
701
00:44:21 --> 00:44:22
No surprise.
702
00:44:22 --> 00:44:27
That's the vector
that we know is in the
703
00:44:27 --> 00:44:36
everything shifting the same
amount, nothing stretching.
704
00:44:36 --> 00:44:40
Talking fast here about
positive definite matrices.
705
00:44:40 --> 00:44:42
This is the key.
706
00:44:42 --> 00:44:44
Let's just ask a few questions
about positive definite
707
00:44:44 --> 00:44:49
matrices as a way to practice.
708
00:44:49 --> 00:44:50
Suppose I had one.
709
00:44:50 --> 00:44:52
Positive definite.
710
00:44:52 --> 00:44:57
What about its inverse?
711
00:44:57 --> 00:45:02
Is that positive
definite or not?
712
00:45:02 --> 00:45:06
So I've got a positive definite
one, it's not singular, it's
713
00:45:06 --> 00:45:09
got positive eigenvalues,
everything else.
714
00:45:09 --> 00:45:14
It's inverse will be
symmetric, so I'm allowed
715
00:45:14 --> 00:45:16
to think about it.
716
00:45:16 --> 00:45:20
Will it be positive definite?
717
00:45:20 --> 00:45:23
What do you think?
718
00:45:23 --> 00:45:27
Well, you've got a whole
bunch of tests to sort
719
00:45:27 --> 00:45:30
of mentally run through.
720
00:45:30 --> 00:45:35
Pivots of the inverse, you
don't want to touch that stuff.
721
00:45:35 --> 00:45:36
Determinants, no.
722
00:45:36 --> 00:45:39
What about eigenvalues?
723
00:45:39 --> 00:45:42
What would be the eigenvalues
if I have this positive
724
00:45:42 --> 00:45:44
definite symmetric matrix,
its eigenvalues are
725
00:45:44 --> 00:45:46
one, four, five.
726
00:45:46 --> 00:45:49
What can you tell me
about the eigenvalues
727
00:45:49 --> 00:45:53
of the inverse matrix?
728
00:45:53 --> 00:45:54
They're the inverses.
729
00:45:54 --> 00:45:56
So those three eigenvalues are?
730
00:45:56 --> 00:46:00
1, 1/4, 1/5, what's
the conclusion here?
731
00:46:00 --> 00:46:02
It is positive definite.
732
00:46:02 --> 00:46:04
Those are all positive,
it is positive definite.
733
00:46:04 --> 00:46:08
So if I invert a positive
definite matrix, I'm
734
00:46:08 --> 00:46:11
still positive definite.
735
00:46:11 --> 00:46:13
All the tests would
have to pass.
736
00:46:13 --> 00:46:17
It's just I'm looking each
time for the easiest test.
737
00:46:17 --> 00:46:22
Let me look now, for the
easiest test on K_1+K_2.
738
00:46:22 --> 00:46:25
739
00:46:25 --> 00:46:27
Suppose that's positive
definite and that's
740
00:46:27 --> 00:46:29
positive definite.
741
00:46:29 --> 00:46:33
What if I add them?
742
00:46:33 --> 00:46:35
What do you think?
743
00:46:35 --> 00:46:38
Well, we hope so.
744
00:46:38 --> 00:46:42
But we have to say which of my
one, two, three, four, five
745
00:46:42 --> 00:46:45
would be a good way to see it.
746
00:46:45 --> 00:46:48
Would be a good way to see it.
747
00:46:48 --> 00:46:50
Good question.
748
00:46:50 --> 00:46:53
Four?
749
00:46:53 --> 00:46:55
We certainly don't want to
touch pivots and now we
750
00:46:55 --> 00:46:58
don't want to touch
eigenvalues either.
751
00:46:58 --> 00:47:03
Of course, if number four
works, others will also work.
752
00:47:03 --> 00:47:05
The eigenvalues will
come out positive.
753
00:47:05 --> 00:47:08
But not too easy to
say what they are.
754
00:47:08 --> 00:47:14
Let's try test number four.
755
00:47:14 --> 00:47:15
So K_1.
756
00:47:15 --> 00:47:18
757
00:47:18 --> 00:47:20
What's the test?
758
00:47:20 --> 00:47:23
So test number four tells us
that this part, x transpose
759
00:47:23 --> 00:47:28
K_1*x, that that part
is positive, right?
760
00:47:28 --> 00:47:30
That that part is positive.
761
00:47:30 --> 00:47:33
If we know that's
positive definite.
762
00:47:33 --> 00:47:37
Now, about K_2 we also know
that for every x, you see it's
763
00:47:37 --> 00:47:42
for every x, that helps, don't
let me put x_2 there, for every
764
00:47:42 --> 00:47:47
x this will be positive.
765
00:47:47 --> 00:47:52
And now what's the
step I want to take?
766
00:47:52 --> 00:47:57
To get some information
on the matrix K_1+K_2.
767
00:47:57 --> 00:47:59
768
00:47:59 --> 00:48:01
I should add.
769
00:48:01 --> 00:48:07
If I add these guys, you see
that it just, then I can
770
00:48:07 --> 00:48:14
write that as, I can
write that this way.
771
00:48:14 --> 00:48:17
And what have I learned?
772
00:48:17 --> 00:48:19
I've learned that that's
positive, even greater than,
773
00:48:19 --> 00:48:21
except for the zero vector.
774
00:48:21 --> 00:48:23
Because this was greater
than, this is greater than.
775
00:48:23 --> 00:48:27
If I add two positive numbers,
the energies are positive
776
00:48:27 --> 00:48:29
and the energies just add.
777
00:48:29 --> 00:48:34
The energies just add.
778
00:48:34 --> 00:48:40
So that definition four was the
good way, just nice, easy way
779
00:48:40 --> 00:48:44
to see that if I have a couple
of positive definite matrices,
780
00:48:44 --> 00:48:47
a couple of positive energies,
I'm really coupling
781
00:48:47 --> 00:48:49
the two systems.
782
00:48:49 --> 00:48:53
This is associated somehow.
783
00:48:53 --> 00:48:55
I've got two systems, I'm
putting them together
784
00:48:55 --> 00:49:00
and the energy is just
even more positive.
785
00:49:00 --> 00:49:05
It's more positive either of
these guys because I'm adding.
786
00:49:05 --> 00:49:11
As I'm speaking here, will you
allow me to try test number
787
00:49:11 --> 00:49:14
five, this A transpose
A business?
788
00:49:14 --> 00:49:20
Suppose K_1 was A transpose A.
789
00:49:20 --> 00:49:21
If it's positive
definite, it will.
790
00:49:21 --> 00:49:31
Be And suppose K_2
is B transpose B.
791
00:49:31 --> 00:49:33
If it's positive
definite, it will be.
792
00:49:33 --> 00:49:42
Now I would like to write the
sum somehow as, in this
793
00:49:42 --> 00:49:43
something transpose something.
794
00:49:43 --> 00:49:47
And I just do it now because
I think it's like, you
795
00:49:47 --> 00:49:54
won't perhaps have thought
of this way to do it.
796
00:49:54 --> 00:49:56
Watch.
797
00:49:56 --> 00:50:01
Suppose I create
the matrix [A; B].
798
00:50:01 --> 00:50:03
That'll be my new matrix.
799
00:50:03 --> 00:50:08
Say, call it C.
800
00:50:08 --> 00:50:11
Am I allowed to do that?
801
00:50:11 --> 00:50:13
I mean, that creates a matrix?
802
00:50:13 --> 00:50:18
These A and B, they had the
same number of columns, n.
803
00:50:18 --> 00:50:20
So I can put one over the
other and I still have
804
00:50:20 --> 00:50:22
something with n columns.
805
00:50:22 --> 00:50:24
So that's my new matrix C.
806
00:50:24 --> 00:50:26
And now I want C transpose.
807
00:50:26 --> 00:50:31
By the way, I'd call
that a block matrix.
808
00:50:31 --> 00:50:35
You know, instead of numbers,
it's got two blocks in there.
809
00:50:35 --> 00:50:37
Block matrices are
really handy.
810
00:50:37 --> 00:50:43
Now what's the transpose
of that block matrix?
811
00:50:43 --> 00:50:47
You just have faith, just
have faith with blocks.
812
00:50:47 --> 00:50:48
It's just like numbers.
813
00:50:48 --> 00:50:55
If I had a matrix [1; 5] then
I'd get a row one, five.
814
00:50:55 --> 00:50:57
But what do you think?
815
00:50:57 --> 00:51:01
This is worth thinking
about even after class.
816
00:51:01 --> 00:51:05
What would be, if this C matrix
is this block A above B, what
817
00:51:05 --> 00:51:07
do you think for C transpose?
818
00:51:07 --> 00:51:11
A transpose, B transpose
side by side.
819
00:51:11 --> 00:51:15
Just put in numbers
and you'd see it.
820
00:51:15 --> 00:51:19
And now I'm going to take
C transpose times C.
821
00:51:19 --> 00:51:25
I'm calling it C now instead of
A because I've used the A in
822
00:51:25 --> 00:51:27
the first guy and I've used B
in the second one and
823
00:51:27 --> 00:51:31
now I'm ready for C.
824
00:51:31 --> 00:51:35
How do you multiply
block matrices?
825
00:51:35 --> 00:51:37
Again, you just have faith.
826
00:51:37 --> 00:51:39
What do you think?
827
00:51:39 --> 00:51:41
Tell me the answer.
828
00:51:41 --> 00:51:44
A transpose, I multiply
that by that just as
829
00:51:44 --> 00:51:47
if they were numbers.
830
00:51:47 --> 00:51:52
And I add that times that just
as if they were numbers.
831
00:51:52 --> 00:51:55
And what do I have?
832
00:51:55 --> 00:51:55
I've got K_1+K_2.
833
00:51:55 --> 00:51:58
834
00:51:58 --> 00:52:05
So I've written K_1, this is
K_1+K_2 and this is in my form
835
00:52:05 --> 00:52:08
C transpose C that I was
looking for, that number
836
00:52:08 --> 00:52:10
five was looking for.
837
00:52:10 --> 00:52:12
So it's done it.
838
00:52:12 --> 00:52:13
It's done it.
839
00:52:13 --> 00:52:19
The fact of getting A, K_1 in
this form, K_2 in this form.
840
00:52:19 --> 00:52:21
And I just made a block
matrix and I got K_1+K_2.
841
00:52:21 --> 00:52:25
842
00:52:25 --> 00:52:29
That's not a big deal in
itself, but block matrices
843
00:52:29 --> 00:52:32
are really handy.
844
00:52:32 --> 00:52:36
It's good to take that
step with matrices.
845
00:52:36 --> 00:52:40
Think of, possibly, the entries
as coming in blocks and
846
00:52:40 --> 00:52:42
not just one at a time.
847
00:52:42 --> 00:52:44
Well, thank you, okay.
848
00:52:44 --> 00:52:51
I swear Friday we'll start
applications in all kinds of
849
00:52:51 --> 00:52:55
engineering problems and
you'll have new applications.
850
00:52:55 --> 00:52:55 |
More About
This Textbook
Overview
This book argues that the teaching of elementary linear algebra can be made more effective by emphasizing applications, expositions, and pedagogy.
This volume grew out of the work of the Linear Algebra Curriculum Study Group and the 1993 Special issue on Linear Algebra of the College Mathematics Journal.
Included are the recommendations of the Linear Algebra Curriculum Study Group, with their core syllabus for the first course, and the thoughts of mathematics faculty who have taught linear algebra using these recommendations. It includes elucidation of these ideas, trenchant criticism of them, and a report on putting them into practice.
A valuable resource for anyone teaching linear algebra. This book argues that the teaching of elementary linear algebra can be made more effective by emphasizing applications, exposition, and pedagogy.
* Relevant applications serve as motivation for all students and as sources of stimulating and challenging problems.
* Effective exposition that finds the right way to communicate concepts is especially important in the teaching of linear algebra, often the first course in which students come to grips with abstraction and complexity and with multiple representations of the same idea.
* Attention to pedagogy that takes into account how students learn technology, and new teaching ideas such as cooperative learning can go a long way toward improving the teaching of linear algebra. This attention helps to increase student understanding of the material.
Contains a core syllabus for the first course in linear algebra, and the thoughts of mathematics faculty members who have taught linear algebra using these recommendations.
Editorial Reviews
Booknews
Provides college and high school mathematics teachers with some of the core findings from research into reforming how calculus is taught, emphasizing the importance of application, exposition, and pedagogy. Includes recommendation of the Linear Algebra Curriculum Study Group and elucidation, criticism, and advice from teachers who have used them. No index. Annotation c. by Book News, Inc., Portland, Or.
Related Subjects
Read an Excerpt
Introduction
The idea of "calculus reform" goes back to the Conference/Workshop to Develop Alternative Curriculum and Teaching Methods for Calculus at the College Level held at Tulane University in 1986. With this conference, and the volumes "Toward a Lean and Lively Calculus" and "Calculus for a New Century", we were all confronted with a challenge. The way we had taught calculus wasn't working; given today's students, faculty, and technology, how should we change it?
It is not surprising that the negative aspects of our students' learning of calculus would be seen to appear in other mathematics courses: in remedial algebra and trigonometry, in differential equations, in the related field of statistics, and in particular in linear algebra.
It is our hope that this volume will be in a fact a "Resource for Teaching Linear Algebra" for instructors in our field, and help them help their students to learn our subject better. In "Resources", we present material on many different aspects of the teaching of linear algebra. The titles of the part of the volume illustrate the breadth of the issues we deal with: The Role of Linear Algebra, Algebra as Seen from Client Disciplines, The Teaching of Linear Algebra, Linear Algebra Expositions, and Applications of Linear Algebra. In each part, the articles will be discussed briefly in an introduction.
We begin our volume with a brief history of linear algebra curriculum reform in the United States in recent years.
Workshops on Computing in the Teaching of Linear Algebra
The first "teaching reform" activities in linear algebra were workshops, short courses, mini-course, etc. on the use of computing in teaching linear algebra. By now, many people have organized such activities; the list includes Homer Bechtell, Jane Day, Benny Evans, Eugene Herman and Charles Jepsen, David Hill and David Zitarelli, Jerry Johnson, Don LaTorre, Steve Leon, Koo Rijpkema, and Kermit Sigmon. Of these, the most ambitious have been the NSF-sponsored ATLAST Workshops of Steve Leon. Many faculty have taken part in these activities, and are now using computing in their linear algebra classrooms. A number of them have made presentations about their activities at national or regional or special-interest meetings.
This is now a variety of elementary linear algebra textbooks containing computing exercises (see Jane Day's article in this volume). There are also books and other materials specifically dealing with computing in linear algebra.
The Linear Algebra Curriculum Study Group
"Linear algebra reform" activities beyond computing workshops go back to 1989, when the participants at an NSF-sponsored Summer Short course on Matrices (organized at Laramie by Duane Porter; the principal speaker was Charles Johnson) decided to write down issues they saw facing linear algebra instructors. Porter organized a Panel Discussion on Linear Algebra in the Undergraduate Curriculum at the 1990 Joint Mathematics Meetings in Louisville. The panelists were Irving Katz and John Poole from the Summer Short Course, David Carlson, and David Lay. Three hundred people attended, most for the entire three hour session! A lively audience discussion followed the panelists' presentations.
After the session, Carlson, Johnson, Lay and Porter organized the Linear Algebra Curriculum Study group. The first LACSG activity was an NSF-funded Workshop at the College of William and Mary held August 7-11, 1990, which involved a broadly-based panel of 20 people from academia and industry. A series of recommendations, including a core syllabus for the first course in linear algebra, was prepared and disseminated. It is reproduced here.
There have been special sessions at all Joint Mathematical Meetings since 1991: at San Francisco (1991 and 1995), Baltimore (1992), San Antonio (193), Cincinnati (1994), Orlando (1996) and San Diego (19997). These have been organized by David Lay, Don LaTorre, Steve Leon, and Duane Porter. There have been many speakers on aspects of the teaching of linear algebra, and a great deal of audience interest.
Another project of the organizers of the LACSG, also with NSF support, has been to prepare a collection of "Gems of Linear Algebra": especially insightful proofs, longer expositional items, and some problems for students. This is intended to assist instructors in their presentation of fundamental linear algebra ideas in improved ways. This volume is nearing completion at this time.
The Special Issue on Linear Algebra of the College Mathematics Journal
One of the "special" activities in linear algebra curricular discussion was the publication in January 1993 of a Special Issue on Linear Algebra of the College Mathematics Journal, edited at that time by Ann and Bill Watkins. This volume contains a number of articles that appeared first in the Special Issue. It also includes some articles which appeared originally in the American Mathematical Monthly and in other issues of the College Mathematics Journal.
Review (a fair-use review excerpt; the source must be cited):
"It contains much extremely interesting material on specific topics in linear algebra and their teaching (including the use of various packages): so it is, indeed, a resource for the teaching of linear algebra…. This rich and fascinating book will help you respond creatively to it….Every lecturer teaching linear algebra should have a copy." - Philip Mahar, Middlesex University, Queensway, Enfield: Mathematical Gazette
"This book argues that the teaching of elementary linear algebra can be made more effective by emphasizing applications, exposition and pedagogy. Relevant applications serve as motivation for all students and as sources of stimulating and challenging problems. Effective exposition that finds the right way to communicate concepts is especially important in teaching of linear algebra, often the first course in which students come to grips with abstraction and complexity, and with multiple representations of the same idea. Attention to pedagogy that takes into account how students learn, technology, and new teaching ideas such as cooperative learning can go a long way toward improving the teaching of linear algebra. " Zentrallblat fur Mathematik
Part III The Teaching of Linear Algebra
Teaching Linear Algebra: Must the Fog Always Roll In?: David Carlson, Dan Diego State University
The Linear Algebra Curriculum Study Group Recommendations for the First course in Linear Algebra: David Carlson, Charles R. Johnson, David C. Lay, A, Duane Porter
A Project on Circles in Space: Carl C. Cowen: Purdue University
Teaching Linear Algebra New Ways: Jane M. Day, San Jose State University
Some Thought on a First Course in Linear Algebra at the College Level: Ed Dubinsky, Purdue University, Purdue University and Education Development Center
The Linear Algebra Curriculum Study Group Recommendations: Moving Beyond Concept Definition; Guershon Harel, Purdue University
Gaussian Elimination in Integer Arithmetic: An Application of the L-U Factorization; Thomas Hern, Bowling Green State University
Iterative Methods in Introductory Linear Algebra; Donald R. LaTorre, Clemson University
Reflections (1988); Robert Mena, California State University Long Beach
Writing About Linear Algebra: Report on an Experiment; Gerald J. Porter
Scenes From Linear Algebra Classes; Shlomo Vitar, the Hebrew University of Jerusalem, Givat Ram
Part IV Linear Algebra Exposition
Down with Determinants!; Sheldon Axler, Michigan State University
Subspaces and Echelon Forms; David C. Lay, University of Maryland
A Geometric Interpretation of the Columns of the (Pseudo)Inverse of A; Melvin J. Maron, University of Louisville, Ghansham M. Manwani, Universidade do Amazonas, Brazil
The Fundamental Theorem of Linear Algebra: Gilbert Strang, Massachusetts Institute of Technology |
Basic Mathematics - 7th edition
Summary: Patient and clear in his explanations and problems, Pat McKeague helps students develop a thorough understanding of the concepts essential to their success in mathematics. Each chapter opens with a real-world application. McKeague builds from the chapter-opening applications, such as the average amount of caffeine in different beverages, and uses the application as a common thread to introduce new concepts, making the material more accessible and engaging for student...show mores. Diagrams, charts, and graphs are emphasized to help students understand the material covered in visual form. McKeague's unique and successful EPAS system of Example, Practice, Answer, and Solution actively involves students with the material and thoroughly prepares them for working the Problem Sets. The Sixth Edition of BASIC MATHEMATICS also features a robust suite of online course management, testing, and tutorial resources for instructors and students. This includes iLrn Testing and Tutorial, vMentor live online tutoring, the Digital Video Companion CD-ROM with MathCue, a Book Companion Web Site featuring online graphing calculator resources, and The Learning Equation (TLE), powered by iLrn. TLE provides a complete courseware package, featuring a diagnostic tool that gives instructors the capability to create individualized study plans. With TLE, a cohesive, focused study plan can be put together to help each student succeed in math Title. We're a Power Distributor; Your satisfaction is our guarantee!
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To present topics demonstrating the beauty and utility of mathematics to the general student population and to provide...
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To present topics demonstrating the beauty and utility of mathematics to the general student population and to provide knowledge and skills useful for college, life and career. The course will include topics related to patterns and reasoning, growth and symmetry, linear and exponential growth, and
Author:
Tim Dube
Date Added:
Jan 15, 2008 |
0521294924
9780521294928
An Introduction to Computational Combinatorics:By the time students have done some programming in one or two languages and have learnt the common ways of representing information in a computer, they will want to embark upon further study of theoretical or applied topics in computer science. Most will encounter problems that require for their solution one or more of the techniques described in this book: for example problems depending upon the formation and solution of different equations; the task of making lists of possible alternatives and of answering questions about them; or the search for discrete optima. Written by the same authors as the highly successful Information Representation and Manipulation in a Computer, this book describes algorithms of mathematical methods and illustrates their application with examples. The mathematical background needed is elementary algebra and calculus. Numerous exercises are provided, with hints to their solutions.
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Rent An Introduction to Computational Combinatorics 1st edition today, or search our site for E. S. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Cambridge University Press. |
TriAGe: Preparation for University Calculus
A refresher of material covered in the Grade 12 Advanced Functions course, TriAGe will help prepare students who plan to enrol in an entry level university calculus course by reinforcing fundamental skills in Trigonometry, Algebra and Geometry. Topics discussed will include: algebraic manipulations used to simplify expressions and solve equations and inequalities; analytic geometry; and polynomial, rational, exponential, logarithmic and trigonometric functions.
Also integrated with the course content will be discussion of specific learning strategies to help students with the transition from high school mathematics to university level expectations.
A final assessment will be provided at the end of the course to students who complete a set of assignments and an end of course evaluation.
Review and deepen your understanding of core concepts to build a strong foundation and increase your chances of success in calculus!
TriAGe will be offered during the Fall term as a blended course. The content will be delivered online, along with a set of online quizzes & practice assignments; students work through the material at their own pace with guidelines provided as to when each unit should be completed.
The course will be augmented by optional weekly tutorials (taking place on the Waterloo campus), at which students have the opportunity to review key concepts and ask questions. Tutorials will be scheduled on:
Wednesdays, Sept 25th to Nov 20th, from 5:00pm - 6:50pm.
Participants can also complete comprehensive assignments for additional feedback on their understanding of concepts. And there will be an option of completing a Final Evaluation at the end of the course (Wed Nov 27th).
Online access to the course will commence on September 9th, with registrations being accepted until 4pm on Monday September 23rd.
Registration Process: To enroll in TriAGe, please complete the Registration Form. If you have any questions about registering, please contact the Office of Continuing Studies: email them at continuingstudies@wlu.ca or call 519-884-0710 ext. 6036. Non-WLU students are welcome! Please take note of the Withdrawal Policy stated below.
**The Department of Mathematics at WLU is permitting successful completion of the TriAGe course to be considered as an equivalency to the Grade 12 Advanced Functions prerequisite for MA100, MA129 and MA110*. To be granted the equivalency, a final grade of 70% or better must be attained through completion of course assignments and the Final Evaluation.
Withdrawal Policy:
Notice of withdrawal must be made in writing to the Office of Continuing Studies;
A full tuition refund, less 10 per cent, will be issued when notice of withdrawal is received up to two weeks prior to the start of the first session;
A full tuition refund, less 25 per cent, will be issued when withdrawal notice is received after two weeks prior to start of the first session and prior to the start of the second session;
After the second sessoin, no refunds will be issued.
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Calculus: Early Transcendentals, 10th Editioncontinues to evolve to fulfill the needs of a changing market by providing flexible solutions to teaching and learning needs of all kinds. Calculus: Early Transcendentals, 10th Editionexcels in increasing student comprehension and conceptual understanding of the mathematics. The new edition retains the strengths of earlier editions: e.g., Anton's trademark clarity of exposition; sound mathematics; excellent exercises and examples; and appropriate level, while incorporating more skill and drill problems within WileyPLUS.
The seamless integration of Howard Anton's Calculus: Early Transcendentals, 10th Edition with WileyPLUS, a research-based, online environment for effective teaching and learning, continues Anton's vision of building student confidence in mathematics because it takes the guesswork out of studying by providing them with a clear roadmap: what to do, how to do it, and whether they did it right.
Readability Balanced with Rigor: The authors' goal is to present precise mathematics to the fullest extent possible in an introductory treatment.
Commitment to Student Success: Clear writing, effective pedagogy--including special exercises designed for self-assessment--and visual representations of the mathematics help students from a variety of backgrounds to learn. Recognizing variations in learning styles, the authors take a "rule of four" approach, presenting concepts from the verbal, algebraic, visual, and numerical points of view to foster deeper understanding whenever appropriate.
Flexibility: This edition is designed to serve a broad spectrum of calculus philosophies-from traditional to "reform." Technology can be emphasized or not, and the order of many topics can be adapted to accommodate each instructor's specific needs.
Quick Check Exercises: Each exercise set begins with approximately five exercises (answers included) that are designed to provide the student with an immediate assessment of whether he or she has mastered key ideas from the section. They require a minimum of computation and can usually be answered by filling in the blanks.
Focus on Concepts Exercises: Each exercise set contains a clearly-identified group of problems that focus on the main ideas of the section.
Technology Exercises: Most sections include exercises that are designed to be solved using either a graphing calculator or a computer algebra system such as Mathematica, Maple, or Derive. These exercises are marked with an icon for easy identification.
Expository Excellence: Clear explanations allow students to build confidence and provide flexibility for the instructor to use class time for problem solving, applications and explanation of difficult concepts.
Mathematical Level: The book is written at a mathematical level that is suitable for students planning on careers in engineering or science.
Applicability of Calculus: One of the primary goals of this text is to link calculus to the real world and the student s own experience. This theme is carried through in the examples and exercises.
Historical Notes: The biographies and historical notes have been a hallmark of this text from its first edition and have been maintained in this edition. All of the biographical materials have been distilled from standard sources with the goal of capturing the personalities of the great mathematicians and bringing them to life for the student. Solutions Manual, Vol II Manual
The Instructor's Manual suggests time allocations and teaching plans for each section in the text. Most of the teaching plans contain a bulleted list of key points to emphasize. The discussion of each section concludes with a sample homework assignment. The Instructor's Manual is also available in PDF format on the password-protected Instructor Companion Site and to users of WileyPLUS.
Web Projects
The Web Projects (Expanding the Calculus Horizon) referenced in the text can also be downloaded from the companion web sites and from WileyPLUS.
Interactive Illustrations
Interactive Illustrations can be used the classroom or computer lab to present and explore key ideas graphically and dynamically. They are especially useful for display of three-dimensional graphs in multivariable calculus. These can be accessed on the Instructor Companion Site and through WileyPLUS.
Printed/Computerized Test Bank
features questions and answers for every section of the text.
PowerPoint lecture slides
cover the major concepts and themes of each section of the book.
ENHANCE YOUR COURSE
A research-based online environment for learning and assessment. Learn more
Student Study Guide
The Student Study Guide is available for download from the Book Companion Site and to users of WileyPLUS.
Web Projects
The Web Projects (Expanding the Calculus Horizon) referenced in the text can also be downloaded from the companion web sites and from WileyPLUS. |
Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger (Dover Books on Mathematics)
Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics—algebraic geometry, in particular. This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Galois theory, Noetherian rings and modules, and rings of fractions. It covers the basics, starting with the divisibility theory in principal ideal domains and ending with the unit theorem, finiteness of the class number, and the more elementary theorems of Hilbert ramification theory. Numerous examples, applications, and exercises appear throughout the text.
Customer Reviews:
Crystal clear
By Smeets Arne - February 23, 2010
Samuel's book is a classic. It is a bit "antique", certainly not the most modern introduction to algebraic number theory. The topics covered in the book are algebraic and integral extensions, Dedekind rings, ideal classes and Dirichlet's unit theorem, the splitting of primes in an extension field and some Galois theory for number fields. So the range of topics is quite small (and the book is short, ~100 pages).
And yet I love this book. It is crystal clear, well written and well structured; it is quite dense (especially the last chapter) and makes the beginning student of algebraic number theory think a lot, but without ever getting too heavy to digest. The list of problems is fantastic: there are many very concrete problems which sharpen your understanding of the material considerably. And last but not least, it is ridiculously cheap.
I recommend this to anyone who wants to learn the basic material about number fields. Without any hesitation.
beautifully written
By Narada "the wondering jew" - December 27, 2010
(I have read this book in French, so can't comment on the translation). This is the best introduction to number theory, and the Dover price certainly can't be beat. The one problem with this book is that the perspective is very Bourbakist, and so the arguments do not tend to be constructive, so you might want your next book to be
Dense and hard to read
By Ben Kraft - April 6, 2013
This book is fairly classic, but it's pretty dense and hard to read. The typesetting is also pretty bad. You're probably better off using one of the various good sets of online notes for algebraic number theory (J.S. Milne's, for example). If you want a book, though, this one will do, and it's nice and small.
The 18th century was a wealth of knowledge, exploration and rapidly growing technology and expanding record-keeping made possible by advances in the printing press. In its determination to preserve ... |
Reinforced Concrete Beam Design and Analysis. Edition No. 1
In this book MathCAD work sheet will be used as a teaching and learning tool for the design and analysis of reinforced concrete beam using ACI code of design. The book has two sample problems that analyze and design both regular and irregular beams. First sample problems explain the use of MathCAD in design and analysis of simple rectangular beam. The 2nd sample problem explain the use of MathCAD in design and analysis of single and double reinforced regular beams such as rectangular beams and irregular beams such as T-beam, L-beam, I -beam and other shapes. In addition to bending reinforcement calculation, the work sheet will calculate the spacing requirement foe shear reinforcement and it will check beam adequacy for the shear. This MathCAD work sheet is intended for civil engineering students and professionals. Civil engineering students can use the work sheet to solve and understand the design and analysis of regular and irregular beams. This worksheet will be followed by other worksheets such as: Design of one way and two way slabs, design of axial and biaxial columns, and design of square, rectangular, and combined foundation.
Mohammed, AL- Ansari. Mohammed Bin Salem is currently an Associate Professor in the Civil Engineering department at the University of Qatar. In 1992, he received his Ph.D. in civil engineering from Catholic University of America. His research interests include earthquake,design of concrete and steel structures. MathCAD applications in analysis and |
1957X / ISBN-13: 9780073519579 ...Show more college math courses and their future teaching experiences, along with helpful ideas for presenting math to their students in a way that will generate interest and enthusiasm. The text draws heavily on NCTM Standards and contains many pedagogical elements designed to foster reasoning, problem-solving and communication skills. The ninth edition represents a significant step forward in terms of online course management as roughly half of all problems in the text will be assignable through our new online homework platform, Connect Mathematics. In addition, Connect Mathematics will be fully integrated with Blackboard, providing the deepest integration of an online homework and course management system in the market today. Additionally, this text can be packaged with an activity set that corresponds to each section of the companion text, " Mathematics for Elementary Teachers: An Activity Approach, " also by the Bennett, Burton, and Nelson team. " Mathematics for Elementary Teachers: An Activity Approach " can be used independently or along with its companion, " Mathematics for Elementary Teachers: A Conceptual Approach. "Hide |
COURSE REQUIREMENTS:
Be prepared – Come to class with all of required materials.
Be on time – in your seat, working on the quiz when the class starts.
Be respectful of everyone and everything in the class.
Always try your hardest – Ask questions if you don't understand.
Teach someone – research has shown that your knowledge and retention of mathematics
goes up exponentially if you help someone else.
INSTRUCTORS' COMMENT TO STUDENTS:
This is a "roll up your sleeves and get to work" class. We work in groups. Lecture is held to a minimum. If you have a question, ask your partners first, and then if you are unsatisfied ask one of your resident staff or teacher. Research shows that students do better in algebra when this format is used. |
Mathematics Syllabus for West Bengal Joint Entrance Examination 2013
West Bengal Joint Entrance Examination Board releases WBJEE 2013 Notification for admission to undergraduate degree level Engineering & Technology courses including Pharmacy and Architecture in different Universities / Government Colleges as well as Self Financing Institutes of West Bengal.
Quadratic Equations: Quadratic equations with real coefficients; Relations between roots and coefficients; Nature of roots; Formation of a quadratic equation, sign and magnitude of the quadratic expression ax2+bx+c (where a, b, c are rational numbers and a ≠ 0).
Permutation and combination: Permutation of n different things taken r at a time (r ≤ n). Permutation of n things not all different. Permutation with repetitions (circular permutation excluded).Combinations of n different things taken r at a time (r ≤ n). Combination of n things not all different. Basic properties.Problems involving both permutations and combinations.
Principle of mathematical induction: Statement of the principle, proof by induction for the sum of squares, sum of cubes of first n natural numbers, divisibility properties like 22n— 1 is divisible by 3 (n ≥ 1), 7 divides 3 2n+1+2n+2(n≥ 1)
Matrices: Concepts of m x n (m ≤ 3, n ≤ 3) real matrices, operations of addition, scalar multiplication and multiplication of matrices. Transpose of a matrix. Determinant of a square matrix. Properties of determinants (statement only). Minor, cofactor and adjoint of a matrix. Non singular matrix. Inverse of a matrix. Finding area of a triangle. Solutions of system of linear equations. (Not more than 3 variables).
Sets, Relations and Mappings: Idea of sets, subsets, power set, complement, union, intersection and difference of sets, Venn diagram, De Morgan's Laws, Inclusion / Exclusion formula for two or three finite sets,Cartesian product of sets.Relation and its properties. Equivalence relation — definition and elementary examples, mappings, range and domain, injective, surjective and bijective mappings, composition of mappings, inverse of a mapping.
Coordinate geometry of two dimensions Basic Ideas: Distance formula, section formula, area of a triangle, condition of collinearity of three points in aplane. Polar coordinates, transformation from Cartesian to polar coordinates and vice versa. Parallel transformation of axes, concept of locus, elementary locus problems.
Straight line: Slope of a line. Equation of lines in different forms, angle between two lines. Condition of perpendicularity and parallelism of two lines. Distance of a point from a line. Distance between two parallel lines. Lines through the point of intersection of two lines.
Circle: Equation of a circle with a given center and radius. Condition that a general equation of second degree in x, y may represent a circle. Equation of a circle in terms of endpoints of a diameter . Parametric equation of a circle. Intersection of a line with a circle. Equation of common chord of two intersecting circles.
Parabola : Standard equation. Reduction of the form x = ay2+by+c or y = ax2+bx+c to the standard form y2= 4ax or x2= 4ay respectively. Elementary properties and parametric equation of a parabola.
Ellipse and Hyperbola: Reduction to standard form of general equation of second degree when xy term isabsent. Conjugate hyperbola. Simple properties. Parametric equations. Location of a point with respect toa conic.
Integral calculus: Integration as a reverse process of differentiation, indefinite integral of standard functions.Integration by parts. Integration by substitution and partial fraction.Definite integral as a limit of a sum with equal subdivisions. Fundamental theorem of integral calculus and its applications. Properties of definite integrals.
Application of Calculus: Tangents and normals, conditions of tangency. Determination of monotonicity, maxima and minima. Differential coefficient as a measure of rate.Motion in a straight line with constant acceleration.Geometric interpretation of definite integral as area, calculation of area bounded by elementary curves and Straight lines. Area of the region included between two elementary curves.
Candidates are informed to prepare the above mentioned syllabus of Mathematics to secure more marks to get Better Admissions in institutions of West Bengal. |
The Train the Trainer Guide: Health Disparities Education, developed by the Society of General Internal Medicine (SGIM)...
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The Train the Trainer Guide: Health Disparities Education, developed by the Society of General Internal Medicine (SGIM) Disparities Task Force (DTF) is a comprehensive tool to facilitate developing, implementing and evaluating health disparities education. The Guide includes five modules highlighting several fundamental concepts in health disparities, suggestions for teaching about health disparities in a wide range of settings and strategies for curriculum evaluation at various stages of training (undergraduate, graduate, and continuing medical education). The modules include: Disparities Foundations, Teaching Disparities in the Clinical Setting, Disparities Beyond the Clinical Setting, Teaching about Disparities Through Community Involvement, and Curriculum Evaluation'Taking geometry this year? Isosceles is here to help!Fresh and easy to use, Isosceles is the perfect drawing tool for...
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'Taking geometry this year? Isosceles is here to help!Fresh and easy to use, Isosceles is the perfect drawing tool for geometry students and teachers.FEATURES:Simple, versatile drawing tools• Includes tools for creating lines, circles, and polygons.• The compass tool lets you create myriad constructions from angle bisectors to spirals.• Isosceles automatically snaps new additions to nearby objects, keeping your drawing accurate so you can focus on the construction.• Draw perpendicular bisectors, angle bisectors, altitudes, and other constructions that automatically stay snapped together.• Toggle between heavy and light pencils to bring out the important parts of the drawing. Change the stroke and fill color of figures.• Add text annotations anywhere in the drawing.• Three distinct modes (Classical, Cartesian, and Isometric) that change settings quickly and easily to fit any kind of drawing.Measure objects easily• View and edit information about any object, such as the length of a line or the circumference of a circle.• Add marks to show congruency and parallel objects, extend lines, show live measurements, and more.Tools for students and teachers• Isosceles features a handy notes sidebar so students can type notes during class without leaving the app. Teachers could use this feature to jot down notes to introduce while teaching.• Connect your iPhone (4 or later) or iPad to a TV or projector to present class material.• Put together homework assignments quickly by constructing diagrams with Isosceles.'This app costs $2.99
״'Meet the Insects: Village Edition' invites everyone to explore high quality photos and videos of insects commonly found in...
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״'Meet the Insects: Village Edition' invites everyone to explore high quality photos and videos of insects commonly found in the village and to enjoy full new narration that will breathe life into all the descriptions.״'+ Tips to become an insect expert: 1. Take a listen to all the audio-narrated insect descriptions and video commentaries!2. While switching back and forth from day and night mode, touch and tap all the insects near the village to watch them wiggle.3. Cicadas? Files? Ladybugs? Grasshoppers? Choose your favorite insects and meet them through close-up videos and mind-blowing photos.4. How do grasshoppers and cicadas produce sounds? Watch and interact with the animation as it takes you into the world of insects.5. Why do flies keep rubbing their legs? Get ready to dive into a handful of interesting facts of each insect with eye-catching illustrations.6. If you think you are done with the encyclopedia, it's time for the OX and Photo quiz!7. Once you find an insect around your house to start the observation journal, you are now the real insect expert!'This app costs $3.99
'Ansel & Clair series (winner of 15 awards) launches second app in its dinosaur trilogy, Ansel and Clair: Jurassic...
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'Ansel & Clair series (winner of 15 awards) launches second app in its dinosaur trilogy, Ansel and Clair: Jurassic Dinosaurs (video demo at Spectacular interactions & animations, stunning visuals, puzzles & games bring Dinosaurs to life▪ Allosaurus, a giant meat-eating dinosaur that was an ancestor of the Tyrannosaurus Rex▪ Apatosaurus, a large Sauropod that was over 80 feet long and weighed more than 5 elephants▪ Archaeopteryx, the earliest known flying dinosaur, which had both feathers and scales▪ Hybodus, which was a shark-like water animal that lived during all three dinosaur periods▪ Brachiosaurus or Arm Lizard, a gigantic Sauropod, which had longer fore limbs than legs▪ Plesiosaurus, a marine reptile that did not lay eggs, but gave birth to live young in the water▪ Stegosaurus, a popular armored plant-eating dinosaur ★ Science, history, geography and geology incorporated in a fun, contextual manner▪ Why did the Apatosaurus swallow stones?▪ Did modern birds evolve from the flying dinosaurs?▪ Why were there more plant-eating dinosaurs than meat-eating dinosaurs?▪ Were there polar ice caps during the Jurassic? ▪ How can you tell the age of a fossil? How do you know where to look for fossils?▪ Why are you more likely to find fossils in sedimentary rocks and not in igneous or metamorphic rocks?▪ How did new pine cones grow if there were no pine flowers?★ Much loved features from other Ansel & Clair apps◆ Touch and drag camera to take photos of dinosaurs and other animals of the Jurassic period ◆ Collect and arrange photos in the Travel Log, and even type your own notes'This app costs $1.99 |
A Problem Solving Approach to Mathematics for Elementary School. Students using this text will receive solid preparation in mathematics, devel... MOREop confidence in their math skills and benefit from teaching and learning techniques that really work. Setting the Standard for Tomorrow' s Teachers: This best-selling text continues as a comprehensive, skills-based resource for future teachers. In this edition, students will benefit from additional emphasis on active and collaborative learning. Revised and updated content will better prepare your students for the day when they will be teachers with students of their own. |
This course is a review of elementary algebra. Topics include real numbers, exponents, polynomials, equation solving and factoring.
†
MATH 0099: Intermediate Algebra
4-0-4. Prerequisite: Satisfactory placement scores/MATH 0097
This course is a review of intermediate algebra. Topics include numbers, linear equations and inequalities, quadratic equations, polynomials and rational expressions and roots. Students must pass the class with a C or better and pass the statewide exit examination.
†
MATH 0099C: Intermediate Algebra Co-requisite
2-0-2. Prerequisite:† COMPASS score of 37 or higher and permission of instructor
This course is a supplement to Math 1111 and designated as a support to students taking College Algebra concurrently.† Topics covered will be prerequisites to MATH 1111 taken on an as-needed basis and embedded into College Algebra material.† The course content will focus on developing mathematical maturity through conceptual understanding and mastery of foundational skills.† For students in MATH 0099 C with COMPASS placement math scores below 40, attendance in Math 1111 is mandatory; after three absences, the student will be automatically dropped from the course and receive a grade of W or WF.
This course places quantitative skills and reasoning in the context of experiences that students will be likely to encounter. It emphasizes processing information in context from a variety of representations, understanding of both the information and the processing and understanding which conclusions can be reasonably determined. Topics covered include sets and set operations, logic, basic probability, data analysis, linear models, quadratic models and exponential and logarithmic models. This course is an alternative in area A of the core curriculum and is not intended to1: College Algebra
3-0-3. Prerequisite: Satisfactory placement scores/MATH 0099
This course is a functional approach to algebra that incorporates the use of appropriate technology. Emphasis will be placed on the study of functions and their graphs, inequalities, and linear, quadratic, piece-wise defined, rational, polynomial, exponential and logarithmic functions. Appropriate applications will be included. This course is an alternative in Area A of the core curriculum and does3: Precalculus
3-0-3. Prerequisite: MATH 1111 with a grade of C or better
This course is designed to prepare students for calculus, physics and related technical subjects. Topics include an intensive study of algebraic and trigonometric functions accompanied by analytic geometry as well as DeMoivreís theorem, polar coordinates and conic sections. Appropriate technology is utilized in the instructional process.
†
MATH 2008: Foundations of Numbers and Operations
3-0-3. Prerequisite: Math 1001, Math 1101, Math 1111, or Math 1113
This course is an Area F introductory mathematics course for early childhood education majors. This course will emphasize the understanding and use of the major concepts of number and operations. As a general theme, strategies of problem solving will be used and discussed in the context of various topics.
†
MATH 2040: Applied Calculus
3-0-3. Prerequisite:† MATH 1111 with a grade of C or better
Differential and integral calculus of algebraic, logarithmic, and exponential functions, applications to social sciences, business, and economics, such as maximum-minimum problems, marginal analysis, and exponential growth models.† This course is designed for those students for whom the standard calculus sequence is not required.
†
MATH 2200: Elementary Statistics
3-0-3. Prerequisites: MATH 1001/MATH 1111
This is a basic course in statistics at a level that does not require knowledge of calculus. Statistical techniques needed for research in many different fields are presented. Course content includes descriptive statistics, probability theory, hypothesis testing, ANOVA, Chi-square, regression and correlation.
Conic sections, translation and rotation of axes, polar coordinates, parametric equations, vectors in the plane and in three-space, the cross product, cylindrical and spherical coordinates, surfaces in three-space, vector fields, line and surface integrals, Stokeís theorem, Greenís theorem and differential equations are studied in this course.
†
MATH 2280: Discrete Mathematics
3-0-3. Prerequisite: MATH 1113 with a grade of C or better or permission of the instructor or permission of the academic dean.
Special interest courses, which may not be transferable, are offered in response to student demand and interest.† In these courses, through oral or written communication, students will demonstrate the ability to synthesize information and articulate knowledge on issues relating to culture, society, creative expression, or the human experience.
Two topics have been used in previous offerings:† Statistical Thinking in Sports and Ethnomathematics
†
MLCS 0099:† Mathematical Literacy for College Students
4-0-4.† Prerequisite:† COMPASS math score of 20 or higher
This course integrates numeracy, proportional reasoning, algebraic reasoning and understanding of functions. Students will develop conceptual and procedural tools that support the use of key mathematical concepts in a variety of contexts. This course prepares students requiring Learning Support (or otherwise advised to refresh their Mathematical Literacy) to take MATH 1001 (Quantitative Skills and Reasoning).
NOTE:† This course is designed for non-STEM (science, technology, engineering, mathematics) majors.† MLCS 0099 is not intended to prepare students for MATH 1111. Students needing MATH 1111 are advised to enroll in MATH 0097 or MATH 0099 as placement tests indicate.
†
MLCS 0099C:Mathematical Literacy for College Students Co-requisite
2-0-2.† Prerequisites:† COMPASS score of 37 or higher and permission of instructor.
This course is a supplement to Math 1001 and designated as a support to students taking Quantitative Skills and Reasoning concurrently.† Topics covered will be prerequisites to MATH 1001 taken on an as-needed basis and embedded into Quantitative Skills and Reasoning material.† The course content will focus on developing mathematical maturity through conceptual understanding and mastery of foundational skills.† For students in MLCS 0099 CO with COMPASS placement math scores below 40, attendance in Math 1001 is mandatory; after three absences, the student will be automatically dropped from the course and receive a grade of W or WF. |
Intermediate Algebra Concepts and Applications
9780201708486
0201708485
Summary: The Sixth Edition of Intermediate Algebra: Concepts and Applications continues to bring your students a best-selling text that incorporates the five-step problem-solving process, real-world applications, proven pedagogy, and an accessible writing style. The Bittinger/Ellenbogen hardback series has consistently provided teachers and students with the tools needed to succeed in developmental mathematics. With this revi...sion, the authors have maintained all the hallmark features that have made this series so successful, including its five-step problem-solving process, student-oriented writing style, real-data applications, and wide variety of exercises. Among the features added or revised are new Aha! exercises that encourage students to think before jumping in to solve a problem, 20% new and added real-data applications, and 50% more new Skill Maintenance Exercises. This series not only provides students with the tools necessary to learn and understand math, but also provides them with insights into how math works in the world around them |
Synopses & Reviews
Publisher Comments:
(back cover)
Really. This isn't going to hurt at all . . .
Students discover interesting ways to see patterns in math word problems, and then make the correct computations to find solutions. In the process, they work with decimals and fractions, compare rates, and work with proportions and percents. The author presents everyday real-world examples which use problem-solving strategies that align with those of the National Council of Teachers of Mathematics. The examples are presented in easy to understand language and formats that help guide students through problems in statistics, probability, and even the fundamentals of algebra. Titles in Barron's Painless Series are specially designed to remove the painful aspects of classroom instruction and transform learning into fun. A final, fascinating chapter helps students search for math word problems and activities available on the Internet.
For Middle School and High School Students
Synopsis:
Synopsis:
"Synopsis"
by Libri,"Synopsis"
by Netread, |
MATH-575
Development Of Mathematics 3 cr
A study of the development of mathematical notation and ideas from prehistoric times to the present, with special emphasis being placed on elementary mathematics through the calculus. The development and historic background of the new math will be included.
Prereq: Consent of instructor.
MATH-580
Patterns Of Problem Solving 3 cr
This course will expose students to a variety of techniques useful in solving mathematics problems. The experiences gained from this course can be applied to problems arising in all fields of mathematics. The student will have the chance to see how some general techniques can be used as tools in many areas. Homework for this course will consist mostly of solving a large number of mathematics problems.
Prereq: MATH-280 or consent of instructor. (Consent will be given to students with substantial interest in problem solving, and adequate preparation.)
MATH-615
Modern Algebra And Number Theory For The Elementary Teacher 3 cr
An introduction to modern algebra with special emphasis on the number systems and algorithms which underlie the mathematics curriculum of the elementary school. Topics include sets, rings, integral domains, rational numbers, real numbers, complex numbers and polynomials. Students may not receive credit for both MATH-615 and MATH-652.
Prereq: MATH-149 and MATH-152.
MATH-616
Geometry For The Elementary Teacher 3 cr
A study of the intuitive, informal geometry of sets of points in space. Topics include elementary constructions, coordinates and graphs, tessellations, transformations, problem solving, and symmetries of polygons and polyhedra.
Prereq: MATH-149 and MATH-152.
MATH-617
Theory Of Numbers 3 cr
A study of the properties of integers, representation of integers in a given base, properties of primes, arithmetic functions, modulo arithmetic. Diophantine equations and quadratic residues. Consideration is also given to some famous problems in number theory.
Prereq: MATH-441/641 and either MATH-355/555 or consent of instructor.
MATH-653
Abstract Algebra 3 cr
This course is a continuation of MATH-452/652 with emphasis on ring and field theory. Topics include a review of group theory, polynomial rings, divisibility in integral domains, vector spaces, extension fields, algebraic extension fields, etc.
Prereq: MATH-355/555 and MATH-452/652.
MATH-659
Partial Differential Equations 3 cr
MATH-664
Advanced Calculus 3 cr
This course presents a rigorous treatment of the differential and integral calculus of single variable functions, convergence theory of numerical sequences and series, uniform convergence theory of sequences and series of functions, metric spaces, function of several real variables, and the inverse function theorem. This course contains a written component.
Prereq: MATH-301.
MATH-671
Numerical Analysis 3 cr
Emphasis on numerical algebra. The problems of linear systems, matrix inversion, the complete and special eigenvalue problems, solutions by exact and iterative methods, orthogonalization, gradient methods. Consideration of stability and elementary error analysis. Extensive use of microcomputers and programs using a high level language such as PASCAL.
Prereq: MATH-171 and MATH-355/555
MATH-690
Workshop 1-3 cr
MATH-694
Seminar 2 cr
MATH-696
Special Studies 1-3 cr
Prereq: Consent of instructor.
MATH-790
Workshop 1-6 cr
MATH-794
Seminar 1-3 cr
MATH-798
Individual Studies 1-3 cr
MATH-799
Thesis Research 1-6 cr
Students must complete a Thesis Proposal Form in the Graduate Studies Office before registering for this course.
COMPUTER SCIENCE (COMPSCI)
COMPSCI-507
Microcomputer Applications 3 cr
This course will treat a variety of applications of microcomputers, as well as their architecture, design and social impact. |
This applet demonstrates an exponential growth model which plots population P_i for i=1 to i=600 given user input for the initial population P_0 and growth rate G. The difference equation used is P_(i... More: lessons, discussions, ratings, reviews,...
This applet demonstrates a logistic growth model which plots population P_i for i = 1 to i = 600 given user input for the initial population P_0, growth rate G and carrying capacity CC. The difference... More: lessons, discussions, ratings, reviews,...
The FTC applet assists students in understanding the concept of the area under a curve. As x is changed, the curve f(x) is drawn and the area between the curve and the x axis is shaded blue. To the ... |
MATH 101
MATHEMATICAL THINKING
Course info & reviews
Presents mathematical topics and applications in a context designed to promote quantitative reasoning and the use of mathematics in solving problems and making decisions. Suitable for majors in humanities, education and others seeking a broad view of mathematics. No background in algebra required. This course earns three GEPs toward Go...
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MATH 10114
3.53846 Stars
14 ratings
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12/22/2013
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12/19/2013
0 StarsSpring 2013-2014 | 0 of 0 people found this review helpful.
Content was difficult. Professor did not explain well.
12/17/2013
2 StarsFall 2013-2014 | 0 of 0 people found this review helpful.
This was not a hard class, but the professor that i had it with did not know how to teach, and therefore the class was pretty confusing. I ended up dropping the class.
12/17/2013
2 StarsFall 2013-2014 | 0 of 0 people found this review helpful.
This class was not a hard class, but the professor that i had it with did not know how to teach and therefore the class was confusing. I ended up dropping the class.
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Q&A
A) You perform an experiment in which you roll a fair die and flip a fair coin. As the result of... Show
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A) You perform an experiment in which you roll a fair die and flip a fair coin. As the result of the experiment, you write down a single
integer which is obtained as follows: If the die comes up with a D and the coin shows a T, you write down D; if the die comes up with a D
and the coin shows a H, you write down 2*D. For example, if you roll a 5 on the die and the coin shows a tail, you write down 5, while if
you roll a 5 on the die and the coin shows a head, you write down 10. How big is the sample space for this experiment?
The yearly rate of return on the Standard & Poor's 500 (an index of 500 large-cap corporations) is approximately normal. From January 1, 1960
through December 31, 2009, the S&P had a mean yearly return of 10.98 percent, with a standard deviation of about 17.46 percent. Take this normal
distribution to be the distribution of yearly returns over a long period.
(a) In what interval do the middle 95 percent of all yearly returns lie?
(b) Stocks can go do as well as up. What are the worst 2.5 percent of annual returns?
(c) What is the interval of the middle 50 percent on annual returns on stocks, according to the distribution given in the previous
exercises? (Hint: What two numbers mark off the middle 50 percent of any distribution?)
1-Looking at your trophy collection more closely, you note that you have three ping-pong trophies... Show
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1-Looking at your trophy collection more closely, you note that you have three ping-pong trophies, three volleyball trophies, and two math
trophies. You decide to put them on the shelf in such a way that all the trophies of the same type are together. In how many ways can this
be done?
2- After looking at the shelf for a while, you decide that it would look best if all the volleyball trophies came first, then the math
trophies, and then the ping-pong trophies. In how many ways can this be done?
1- You own a group of ten different pets and you need to pick three of them to send to your cousin M... Show more
1- You own a group of ten different pets and you need to pick three of them to send to your cousin Mort who is currently petless and miserable. In how
many different ways can you select the pets to be sent off?
2- On reflection, since all three pets are to be put in a single cage for shipping, you decide that it would not be a good idea to send both Cutty (a
bobcat) and Cheap (a parakeet) to Mort. With this restriction, in how many different ways can you select the pets to be sent off?
3- How many 4-digit numbers can you build if the number must be odd, odd digits can be repeated, and even digits cannot be repeated?
4-How many 4-digit numbers are there which consist of all different digits? |
Find a Mccook, IL Algebra 2Calculus is necessary for studying the way things change from a mathematical standpoint. It has been founded in the 17th century by Leibniz and Newton and was developed primarily to solve problems in physics. Calculus is divided into Differential Calculus, which studies instantaneous rates of c... |
Tagged Questions
I teach average-level high school students who have not had much beyond Algebra 1. I want to show them why induction makes sense. I want the sort of problem where it is intuitive that a statement is ...
Ok, so I got an answer wrong on my exam because my teacher says that the function $f(x)=\frac{(x+2)x}{x+2}=x$ but I insist that it isn't defined for x=-2. If it was then $\frac{x}{x}=1$ for all reals ...
Ok guys so I've been out of school for eight years, never used algebra again, also I was forcibly removed from school in 9th grade. I need to ask a few questions on Elementary Algebra, what are the () ...
$$mc^2.$$
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Here, one of the expression's factors is $m$. Is there a general name for the factors of anAssuming we don't have a calculator that can do summation notation. My class is not up to summation yet, but I'm asking a question involving this concept because I'm not all that experienced using it. ...
It is a common practice to have students of elementary algebra infer the domain of a function as an exercise. I believe this is contrary to the spirit of the definition of a function as a collection ...
I have a non-mathematician friend who is interested in re-learning algebra. I am more than happy to help, but I am in no position to judge what is a good introductory text --- only to identify when a ... |
MS Math 3.0 Rolls Out This Month
By Michelle Rutledge
05/17/07
##AUTHORSPLIT##<--->
Microsoft recently announced the launch of Math 3.0, a math and science educational tool for students in grade levels 6-12, as well as entry-level college students. The software is designed for use at home, to assist students with math and science concepts and homework, or for visual examples in the classroom.
"Microsoft Math provides a space for nurturing student learning in mathematics with dynamic visualizations. The program provides essential ingredients for classroom environments designed to challenge all students to engage in visual thinking," said Margaret L. Neiss, mathematics education professor at Oregon State University in a prepared statement.
According to Microsoft, the software includes study material for six different math and science subjects. Features in the program include:
Graphing calculator;
Step-by-step math solutions;
Formulas and equation library;
Unit conversion tool;
Triangle solver; and
Handwriting support.
Math 3.0 runs $19.95 for a single license. Volume licensing is available for educational institutions |
-Hill's Top 50 Math Skills for GED Success
From making an appropriate estimate and solving for volume, this distinctive workbook features step-by-step instructions, example questions and an ...Show synopsisFrom making an appropriate estimate and solving for volume, this distinctive workbook features step-by-step instructions, example questions and an explanatory answer key, short concise lessons presented on double-page spreads, and more |
high school students and teachers with an interest in mathematical problem-solving, this volume offers a wealth of nonroutine problems in geometry that stimulate students to explore unfamiliar or little-known aspects of mathematics.Included are nearly 200 problems dealing with congruence and parallelism, the Pythagorean theorem, circles, area relationships, Ptolemy and the cyclic quadrilateral, collinearity and concurrency, and many other subjects. Within each topic, the problems are arranged in approximate order of difficulty. Detailed solutions (as well as hints) are provided for all problems, and specific answers for most.Invaluable as a supplement to a basic geometry textbook, this volume offers both further explorations on specific topics and practice in developing problem-solving techniques. |
0387943919
9780387943916
Mathematics and Politics:This book teaches humanities majors the accessibility and beauty of discrete and deductive mathematics. It assumes no prior knowledge of either college-level mathematics or political science, and could be offered at the freshman or sophomore level. The book devotes one chapter each to a model of escalation, game-theoretic models of international conflict, political power, and social choice. The material progresses in level of difficulty as the book progresses.
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Rent Mathematics and Politics 1st edition today, or search our site for Taylor textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Springer. |
Intermediate Algebra for College Students
9780131400597
ISBN:
0131400592
Edition: 6 Pub Date: 2003 Publisher: Prentice Hall PTR
Summary: Intermediate Algebra for College Students, Sixth Editionby Allen R. Angel will guide you through a practical approach to learning algebra. It was written with you, the student, in mind. For example: In section 1. 1, the author presents Study Skills for Success in Mathematics. Following these study skills will increase your chance of success in this course. Boxes marked Helpful Hints provide important information and ...guide students through difficult areas. Exercises are marked with a CD/Videotape icon to indicate that a specific exercise is worked out in detail on the Lecture Videos making it easy for you to review material presented in class. The Lecture Videos are available digitized on CD-ROM or on VHS tape.
Angel, Allen R. is the author of Intermediate Algebra for College Students, published 2003 under ISBN 9780131400597 and 0131400592. Seventy four Intermediate Algebra for College Students textbooks are available for sale on ValoreBooks.com, seventy two used from the cheapest price of $0.91, or buy new starting at $24.85 |
It seems like you are not the only one encountering this problem. A friend of mine was in the same situation last month. That is when he came across this program known as Algebrator. It is by far the best and cheapest piece of software that can help you with problems on free learn algebra. It won't just find a solution for your problems but also give a step by step explanation of how it arrived at that solution.
Algebrator did help my son to have high grades in Algebra. Good thing we found this amazing software because I believe it did not only help him to have high marks in his homeworks but also helped him in his exams since the program helped in explaining the process of solving the problem by displaying the solution.
Algebrator is the program that I have used through several algebra classes - Basic Math, Basic Math and College Algebra. It is a truly a great piece of algebra software. I remember of going through problems with mixed numbers, dividing fractions and syntehtic division. I would simply type in a problem from the workbook, click on Solve – and step by step solution to my algebra homework. I highly recommend the program. |
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68 of 69 people found the following review helpful
5.0 out of 5 starsGCSE revision...
of each topic. The layout is clear and encouraging and it is easy to plan a scheme of work using it.
I already bought AQA GCSE Maths Higher Module 3, 5 and 1 as well as AQA own Revision book but I would strongly recommend this CGP Workbook for Higher level and it's sufficient for all your revision. In term of clarity and explanation it's much better than AQA books and easy to read and understand even the most complex topics-I would strongly recommend and must have for all Higher aiming for A or B-simply it's a brilliant revision book plus all answers at the back!
Very useful for revising as it has the answers in the back. No explanations though so you would need teaching or another book to understand it. Therefore, only for revision purposes. Graduates from easy to difficult in each section. I have found it very useful and recommend it.
This book is good but it needs to be bought in conjunction with the Higher Revision GCSE Revision by the same author and group (cgp) as it explains the equations and marries along to both books very well.
I bought this book to go with the GCSE revision guide. It was suggested by my daughters maths tutor and complements the revision guide really well! I would recommend both books to aid children studying for their GCSE's |
Mathematicians After 1700 - MAT-913 stories behind mathematical discoveries are fascinating but rarely told. When students learn how persons like themselves have discovered and shaped mathematics, their interest and motivation grows. This course examines the lives and work of great mathematicians who lived after 1700. It is designed to help teachers of grades 3-12 show the human dimension of mathematics.
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"I am very pleased with the whole experience with Fresno Pacific University. I believe they offer quality courses that enable teachers to excel and improve in their teaching methods. The interaction I have had with the professor has been encouraging. I sense that there is a genuine care for me as a student." |
Transformations and Projections in Computer Graphics
Transformations and Projections in Computer Graphics provides a thorough background, discussing the mathematics of perspective in a detailed, yet accessible style. It also reviews nonlinear projections in depth, including fisheye, panorama, and map projections frequently used to enhance digital images.
This book provides a thorough background in these two important topics in graphics. The book introduces perspective in an original way and discusses the mathematics of perspective in detail, in an accessible way. It treats nonlinear projections in depth, including the popular fisheye, panorama, and map projections. Only a basic knowledge of linear algebra, vectors, and matrices is required, as key ideas are introduced slowly, examined and illustrated by figures and examples, and enforced through solved exercises. Topics and Features: - Provides a complete and self-contained presentation of the topic's core concepts, principles, and methods - Features a 12-page color section - Includes a wealth of exercises - Integrates a complementary website that supplies additional auxiliary material Written for computer professionals both within and outside the field of Computer Graphics, this succinct text/reference will prove an essential resource for readers. Also suitable for graduates studying in Computer Graphics and CAD courses.
Table of Contents
Table of Contents
From the contents Transformations.
Introduction.
Two
Dimensional Transformations.
Three
Dimensional Coordinate Systems.
Three
Dimensional Transformations.
Transforming the Coordinate System.
Parallel Projections.
Orthographic Projections.
Axonometric Projections.
Oblique Projections.
Perspective Projection.
One Two Three... Infinity.
History of Perspective.
Perspective in Curved Objects.
The Mathematics of Perspective.
General Perspective.
Transforming The Object.
Viewer At An Arbitrary Location.
Coordinate
Free Approach.
The Viewing Volume.
Stereoscopic Images.
Creating a Stereoscopic Image.
Viewing a Stereoscopic Image.
Nonlinear Projections.
False Perspective.
Fisheye Projection.
Circle Inversion.
Panoramic Projections.
Cylindrical Panoramic Projection.
Spherical Panoramic Projection.
Cubic Panoramic Projection.
Six
Point Perspective.
Other Panoramic Projections.
Panoramic Cameras.
Telescopic Projection.
Microscopic Projection.
Anamorphosis.
Map Projections |
The main goal of this project is to improve student understanding of the geometric nature of multivariable calculus concepts,...
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The main goal of this project is to improve student understanding of the geometric nature of multivariable calculus concepts, i.e., to help them develop accurate geometric intuition about multivariable calculus concepts and the various relationships among them.To accomplish this goal, the project includes four parts:· Creating a Multivariable Calculus Visualization applet using Java and publishing it on a website: web.monroecc.edu/calcNSF· Creating a series of focused applets that demonstrate and explore particular 3D calculus concepts in a more dedicated way.· Developing a series of guided exploration/assessments to be used by students to explore calculus concepts visually on their own.· Dissemination of these materials through presentations and poster sessions at math conferences and through other publications.Intellectual Merit: This project provides dynamic visualization tools that enhance the teaching and learning of multivariable calculus. The visualization applets can be used in a number of ways:- Instructors can use them to visually demonstrate concepts and verify results during lectures.- Students can use them to explore the concepts visually outside of class, either using a guided activity or on their own.- Instructors can use the main applet (CalcPlot3D) to create colorful graphs for visual aids (color overheads), worksheets, notes/handouts, or tests. 3D graphs or 2D contour plots can be copied from the applet and pasted into a word processor like Microsoft Word.- Instructors will be able to use CalcPlot3D to create lecture demonstrations containing particular functions they specify and/or guided explorations for their own students using a scripting feature that is being integrated with this applet.The guided activities created for this project will provide a means for instructors to get their students to use these applets to actively explore and "play" with the calculus concepts.Paul Seeburger, the Principal Investigator (PI) for this grant project, has a lot of experience developing applets to bring calculus concepts to life. He has created 100+ Java applets supporting 5 major calculus textbooks (Anton, Thomas, Varberg, Salas, Hughes-Hallett). These applets essentially make textbook figures come to life. See examples of these applets at Impacts: This project will provide reliable visualization tools for educators to use to enhance their teaching in calculus and also in various Physics/Engineering classes. It is designed to promote student exploration and discovery, providing a way to truly "see" how the concepts work in motion and living color. The applets and support materials will be published and widely disseminated through the web and conference presentations.
An interactive multimedia tutorial for healthcare professionals wishing to refresh math skills and learn how to calculate...
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An interactive multimedia tutorial for healthcare professionals wishing to refresh math skills and learn how to calculate medication dosages and intravenous (IV) rates. The tutorial has three modules. Each module has includes a quiz. The first module covers information about fractions, decimals, ratios, proportions, and percentage. The second module covers information about conversions, medication administration, and dosage calculation. The third module covers information about intraveneous infusions including tubing calculation and intraveneous flow rates. Key words: Medication calculation; Intravenous flow rate calculation; Mathematics principles
From the website: "The Crump Institute for Molecular Imaging brings together faculty, students, and staff with a variety of...
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From the website: "The Crump Institute for Molecular Imaging brings together faculty, students, and staff with a variety of backgrounds - physics, mathematics, engineering, biology, chemistry, and medicine - to pursue innovative technologies and science to accelerate our understanding of biology and medicine.״
The resource contains many Flash physics animations covering topics such as chaos, mechanics, vectors, waves, relativity;...
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The resource contains many Flash physics animations covering topics such as chaos, mechanics, vectors, waves, relativity; includes a tutorial on using Flash with mathematical equations to create controlled animations.
This site contains a collection of fully developed high school curriculum modules that use the Internet in significant ways. ...
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This site contains a collection of fully developed high school curriculum modules that use the Internet in significant ways. There are currently 15 modules in Mathematics and 6 modules in Science; also, there are approximately two dozen additional modules that have been created by instructors and/or Education students.The learning modules here are web-based, technology intensive lessons focusing on mathematics and science in an applied context. They have been developed for teachers, by teachers, aligned with the Illinois State Learning Standards and the National Council for Teachers of Mathematics (NCTM) Standards. Some of the lessons are designed to last over several days, some only for a class period.
The MSTE lessons site contains a collection of excellent high school/lower college division math lessons (and a limited...
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The MSTE lessons site contains a collection of excellent high school/lower college division math lessons (and a limited number of science lessons) that use the Internet in significant ways. The lessons have been developed with descriptions of the problem, connections to standards, examples of use, references, and more. Java source codes are often available. The Office for Mathematics, Science, and Technology Education (MSTE) is a division of the College of Education at the University of Illinois at Urbana-Champaign.
A java application that lets you investigate the problem of creating a phylgenetic tree. there is some explanation of the...
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A java application that lets you investigate the problem of creating a phylgenetic tree. there is some explanation of the problem of drawing phylogenetic trees and a couple of versions of the applet. The applets let you randomly generate trees and also let you infer a phylogenetic tree with a couple of different methods. Number of species, mutation rate and the method for calculating the distance matrix can all be varied. |
Basic College Mathematics
Basic College Mathematics
Basic Mathematics
Basic Mathematics
Basic Mathematics
Basic Mathematics
Basic Mathematics
Basic Mathematics
BASIC MATHEMATICS
Basic Mathematics ( AIE )
Collaborative Learning Activities Manual
MathXL Tutorials on CD for Basic College Mathematics
Student Solutions Manual for Basic College Mathematics
Video Resources on DVD with Chapter Test Prep Videos for Basic College Mathematics
Summary
Worksheets for Classroom or Lab Practice offer extra practice exercises for every section of the text, with ample space for students to show their work. These lab- and classroom-friendly workbooks also list the learning objectives and key vocabulary terms for every text section, along with vocabulary practice problems. |
Geometrica09
Exact Drawing and Geometry
Geometrica is designed to be an encyclopedia of
geometry. Because it uses the symbolic engine of Mathematica,
the drawing is exact, many theorems are integrated, and tests of
validity can be performed. List processing is embedded in most
functions so that complex commands can be performed with concise
statements. Adding legends to a figure is an example of that feature.
Euclidean geometry is thoroughly covered for two and three
dimensions. Whenever possible, a unified design is adopted for 2D and
3D, and plane objects can easily be immersed into space. Geometric
objects are systematically represented by their Cartesian or
parametric forms and defined by a variety of Euclidean functions.
Basic objects such as lines, planes, conics, and quadrics are
manipulated using their Cartesian equations. To simplify these
manipulations, default definitions are available so that an object's
shape can be visualized without using all the equation's coefficients.
For instance, HyperbolicParaboloid[] returns a
quadric that can be immediately visualized using the
function Draw3D.
Differential geometry is treated through the parametric form of an object.
A catalog of curves and surfaces is available. Distances, areas, and volumes are evaluated.
Thick objects are readily obtained using the functions Wall and Pipe.
Geometrica is used by mathematics educators at the high school
and college level, by researchers for drawing exact diagrams of any
complexity, and by graphical designers, architects, and engineers for CAD
applications. |
"deMystified" enough
I bought the book about 1/2 way through my College Algebra class. There are some helpful sections, but didn't have several key things. For example, nothing on Matrices, nothing on the Binomial Theorem.
1 out of 1 people found this review helpful.
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