About The Author Howard Anton obtained his B.A from Lehigh University, his M.A from the University of Illinois, and his Ph.D from the Polytechnic University of Brooklyn, all in mathematics In the early 1960s he worked for Burroughs Corporation and Avco Corporation at Cape Canaveral, Florida, where he was involved with the manned space program In 1968 he joined the Mathematics Department at Drexel University, where he taught full time until 1983 Since then he has devoted the majority of his time to textbook writing and activities for mathematical associations Dr Anton was president of the EPADEL Section of the Mathematical Association of America (MAA), served on the Board of Governors of that organization, and guided the creation of the Student Chapters of the MAA In addition to various pedagogical articles, he has published numerous research papers in functional analysis, approximation theory, and topology He is best known for his textbooks in mathematics, which are among the most widely used in the world There are currently more than 150 versions of his books, including translations into Spanish, Arabic, Portuguese, Italian, Indonesian, French, Japanese, Chinese, Hebrew, and German For relaxation, Dr Anton enjoys travel and photography Copyright © 2010 John Wiley & Sons, Inc All rights reserved Preface This edition of Elementary Linear Algebra gives an introductory treatment of linear algebra that is suitable for a first undergraduate course Its aim is to present the fundamentals of linear algebra in the clearest possible way—sound pedagogy is the main consideration Although calculus is not a prerequisite, there is some optional material that is clearly marked for students with a calculus background If desired, that material can be omitted without loss of continuity Technology is not required to use this text, but for instructors who would like to use MATLAB, Mathematica, Maple, or calculators with linear algebra capabilities, we have posted some supporting material that can be accessed at either of the following Web sites: www.howardanton.com www.wiley.com/college/anton Summary of Changes in this Edition This edition is a major revision of its predecessor In addition to including some new material, some of the old material has been streamlined to ensure that the major topics can all be covered in a standard course These are the most significant changes: • Vectors in 2-space, 3-space, and n-space Chapters and of the previous edition have been combined into a single chapter This has enabled us to eliminate some duplicate exposition and to juxtapose concepts in n-space with those in 2-space and 3-space, thereby conveying more clearly how n-space ideas generalize those already familiar to the student • New Pedagogical Elements Each section now ends with a Concept Review and a Skills mastery that provide the student a convenient reference to the main ideas in that section • New Exercises Many new exercises have been added, including a set of True/False exercises at the end of most sections • Earlier Coverage of Eigenvalues and Eigenvectors The chapter on eigenvalues and eigenvectors, which was Chapter in the previous edition, is Chapter in this edition • Complex Vector Spaces The chapter entitled Complex Vector Spaces in the previous edition has been completely revised The most important ideas are now covered in Section 5.3 and Section 7.5 in the context of matrix diagonalization A brief review of complex numbers is included in the Appendix • Quadratic Forms This material has been extensively rewritten to focus more precisely on the most important ideas • New Chapter on Numerical Methods In the previous edition an assortment of topics appeared in the last chapter That chapter has been replaced by a new chapter that focuses exclusively on numerical methods of linear algebra We achieved this by moving those topics not concerned with numerical methods elsewhere in the text • Singular-Value Decomposition In recognition of its growing importance, a new section on Singular-Value Decomposition has been added to the chapter on numerical methods • Internet Search and the Power Method A new section on the Power Method and its application to Internet search engines has been added to the chapter on numerical methods • Applications There is an expanded version of this text by Howard Anton and Chris Rorres entitled Elementary Linear Algebra: Applications Version, 10th (ISBN 9780470432051), whose purpose is to supplement this version with an extensive body of applications However, to accommodate instructors who asked us to include some applications in this version of the text, we have done so These are generally less detailed than those appearing in the Anton/Rorres text and can be omitted without loss of continuity Hallmark Features • Relationships Among Concepts One of our main pedagogical goals is to convey to the student that linear algebra is a cohesive subject and not simply a collection of isolated definitions and techniques One way in which we this is by using a crescendo of Equivalent Statements theorems that continually revisit relationships among systems of equations, matrices, determinants, vectors, linear transformations, and eigenvalues To get a general sense of how we use this technique see Theorems 1.5.3, 1.6.4, 2.3.8, 4.8.10, 4.10.4 and then Theorem 5.1.6, for example • Smooth Transition to Abstraction Because the transition from Rn to general vector spaces is difficult for many students, considerable effort is devoted to explaining the purpose of abstraction and helping the student to “visualize” abstract ideas by drawing analogies to familiar geometric ideas • Mathematical Precision When reasonable, we try to be mathematically precise In keeping with the level of student audience, proofs are presented in a patient style that is tailored for beginners There is a brief section in the Appendix on how to read proof statements, and there are various exercises in which students are guided through the steps of a proof and asked for justification • Suitability for a Diverse Audience This text is designed to serve the needs of students in engineering, computer science, biology, physics, business, and economics as well as those majoring in mathematics • Historical Notes To give the students a sense of mathematical history and to convey that real people created the mathematical theorems and equations they are studying, we have included numerous Historical Notes that put the topic being studied in historical perspective About the Exercises • Graded Exercise Sets Each exercise set begins with routine drill problems and progresses to problems with more substance • True/False Exercises Most exercise sets end with a set of True/False exercises that are designed to check conceptual understanding and logical reasoning To avoid pure guessing, the students are required to justify their responses in some way • Supplementary Exercise Sets Most chapters end with a set of supplementary exercises that tend to be more challenging and force the student to draw on ideas from the entire chapter rather than a specific section Supplementary Materials for Students • Student Solutions Manual This supplement provides detailed solutions to most theoretical exercises and to at least one nonroutine exercise of every type (ISBN 9780470458228) • Technology Exercises and Data Files The technology exercises that appeared in the previous edition have been moved to the Web site that accompanies this text Those exercises are designed to be solved using MATLAB, Mathematica, or Maple and are accompanied by data files in all three formats The exercises and data can be downloaded from either of the following Web sites www.howardanton.com www.wiley.com/college/anton Supplementary Materials for Instructors • Instructor's Solutions Manual This supplement provides worked-out solutions to most exercises in the text (ISBN 9780470458235) • WileyPLUS™ This is Wiley's proprietary online teaching and learning environment that integrates a digital version of this textbook with instructor and student resources to fit a variety of teaching and learning styles WileyPLUS will help your students master concepts in a rich and structured environment that is available to them 24/7 It will also help you to personalize and manage your course more effectively with student assessments, assignments, grade tracking, and other useful tools • Your students will receive timely access to resources that address their individual needs and will receive immediate feedback and remediation resources when needed • There are also self-assessment tools that are linked to the relevant portions of the text that will enable your students to take control of their own learning and practice • WileyPLUS will help you to identify those students who are falling behind and to intervene in a timely manner without waiting for scheduled office hours More information about WileyPLUS can be obtained from your Wiley representative A Guide for the Instructor Although linear algebra courses vary widely in content and philosophy, most courses fall into two categories —those with about 35–40 lectures and those with about 25–30 lectures Accordingly, we have created long and short templates as possible starting points for constructing a course outline Of course, these are just guides, and you will certainly want to customize them to fit your local interests and requirements Neither of these sample templates includes applications Those can be added, if desired, as time permits Long Template Short Template Chapter 1: Systems of Linear Equations and Matrices lectures lectures Chapter 2: Determinants lectures lectures Long Template Short Template Chapter 3: Euclidean Vector Spaces lectures lectures Chapter 4: General Vector Spaces 10 lectures 10 lectures Chapter 5: Eigenvalues and Eigenvectors lectures lectures Chapter 6: Inner Product Spaces lectures lecture Chapter 7: Diagonalization and Quadratic Forms lectures lectures Chapter 8: Linear Transformations lectures lectures 37 lectures 30 lectures Total: Acknowledgements I would like to express my appreciation to the following people whose helpful guidance has greatly improved the text Reviewers and Contributors Don Allen, Texas A&M University John Alongi, Northwestern University John Beachy, Northern Illinois University Przemslaw Bogacki, Old Dominion University Robert Buchanan, Millersville University of Pennsylvania Ralph Byers, University of Kansas Evangelos A Coutsias, University of New Mexico Joshua Du, Kennesaw State University Fatemeh Emdad, Michigan Technological University Vincent Ervin, Clemson University Anda Gadidov, Kennesaw State University Guillermo Goldsztein, Georgia Institute of Technology Tracy Hamilton, California State University, Sacramento Amanda Hattway, Wentworth Institute of Technology Heather Hulett, University of Wisconsin—La Crosse David Hyeon, Northern Illinois University Matt Insall, Missouri University of Science and Technology Mic Jackson, Earlham College Anton Kaul, California Polytechnic Institute, San Luis Obispo Harihar Khanal, Embry-Riddle University Hendrik Kuiper, Arizona State University Kouok Law, Georgia Perimeter College James McKinney, California State University, Pomona Eric Schmutz, Drexel University Qin Sheng, Baylor University Adam Sikora, State University of New York at Buffalo Allan Silberger, Cleveland State University Dana Williams, Dartmouth College Mathematical Advisors Special thanks are due to a number of talented teachers and mathematicians who provided pedagogical guidance, provided help with answers and exercises, or provided detailed checking or proofreading: John Alongi, Northwestern University Scott Annin, California State University, Fullerton Anton Kaul, California Polytechnic State University Sarah Streett Cindy Trimble, C Trimble and Associates Brad Davis, C Trimble and Associates The Wiley Support Team David Dietz, Senior Acquisitions Editor Jeff Benson, Assistant Editor Pamela Lashbrook, Senior Editorial Assistant Janet Foxman, Production Editor Maddy Lesure, Senior Designer Laurie Rosatone, Vice President and Publisher Sarah Davis, Senior Marketing Manager Diana Smith, Marketing Assistant Melissa Edwards, Media Editor Lisa Sabatini, Media Project Manager Sheena Goldstein, Photo Editor Carol Sawyer, Production Manager Lilian Brady, Copyeditor Special Contributions The talents and dedication of many individuals are required to produce a book such as this, and I am fortunate to have benefited from the expertise of the following people: David Dietz — my editor, for his attention to detail, his sound judgment, and his unwavering faith in me Jeff Benson — my assistant editor, who did an unbelievable job in organizing and coordinating the many threads required to make this edition a reality Carol Sawyer — of The Perfect Proof, who coordinated the myriad of details in the production process It will be a pleasure to finally delete from my computer the hundreds of emails we exchanged in the course of working together on this book Scott Annin — California State University at Fullerton, who critiqued the previous edition and provided valuable ideas on how to improve the text I feel fortunate to have had the benefit of Prof Annin's teaching expertise and insights Dan Kirschenbaum — of The Art of Arlene and Dan Kirschenbaum, whose artistic and technical expertise resolved some difficult and critical illustration issues Bill Tuohy — who read parts of the manuscript and whose critical eye for detail had an important influence on the evolution of the text Pat Anton — who proofread manuscript, when needed, and shouldered the burden of household chores to free up time for me to work on this edition Maddy Lesure — our text and cover designer whose unerring sense of elegant design is apparent in the pages of this book Rena Lam — of Techsetters, Inc., who did an absolutely amazing job of wading through a nightmare of author edits, scribbles, and last-minute changes to produce a beautiful book John Rogosich — of Techsetters, Inc., who skillfully programmed the design elements of the book and resolved numerous thorny typesetting issues Lilian Brady — my copyeditor of many years, whose eye for typography and whose knowledge of language never ceases to amaze me The Wiley Team — There are many other people at Wiley who worked behind the scenes and to whom I owe a debt of gratitude: Laurie Rosatone, Ann Berlin, Dorothy Sinclair, Janet Foxman, Sarah Davis, Harry Nolan, Sheena Goldstein, Melissa Edwards, and Norm Christiansen Thanks to you all Copyright © 2010 John Wiley & Sons, Inc All rights reserved CHAPTER Systems of Linear Equations and Matrices CHAPTER CONTENTS 1.1 Introduction to Systems of Linear Equations 1.2 Gaussian Elimination 1.3 Matrices and Matrix Operations 1.4 Inverses; Algebraic Properties of Matrices 1.5 Elementary Matrices and a Method for Finding 1.6 More on Linear Systems and Invertible Matrices 1.7 Diagonal, Triangular, and Symmetric Matrices 1.8 Applications of Linear Systems • Network Analysis (Traffic Flow) • Electrical Circuits • Balancing Chemical Equations • Polynomial Interpolation 1.9 Leontief Input-Output Models INTRODUCTION Information in science, business, and mathematics is often organized into rows and columns to form rectangular arrays called “matrices” (plural of “matrix”) Matrices often appear as tables of numerical data that arise from physical observations, but they occur in various mathematical contexts as well For example, we will see in this chapter that all of the information required to solve a system of equations such as is embodied in the matrix and that the solution of the system can be obtained by performing appropriate operations on this matrix This is particularly important in developing computer programs for solving systems of equations because computers are well suited for manipulating arrays of numerical information However, matrices are not simply a notational tool for solving systems of equations; they can be viewed as mathematical objects in their own right, and there is a rich and important theory associated with them that has a multitude of practical applications It is the study of matrices and related topics that forms the mathematical field that we call “linear algebra.” In this chapter we will begin our study of matrices Copyright © 2010 John Wiley & Sons, Inc All rights reserved c 13 15 17 19 (b) True/False 9.1 (a) False (b) False (c) True (d) True (e) True Exercise Set 9.2 a dominant b No dominant eigenvalue ; dominant eigenvalue: ; dominant eigenvector: dominant eigenvalue: ; dominant eigenvector: a b c Dominant eigenvalue: ; dominant eigenvector: d 0.1% 13 a Starting with , it takes iterations Starting with , it takes iterations b Exercise Set 9.3 Sites and (tie); sites and are irrelevant Site 2, site 3, site 4; sites and are irrelevant Exercise Set 9.4 a b c , or about 18.5 hours a b c d a b 1334 Exercise Set 9.5 11 True/False 9.5 (a) False (b) True (c) False (d) False (e) True (f) False (g) True Exercise Set 9.6 s for forward phase, 10 s for backward phase 70,100 numbers must be stored; A has 100,000 entries True/False 9.6 (a) True (b) True (c) False Chapter Supplementary Exercises a b c 11 Exercise Set 10.1 a b a or b or (a parabola) a b a b ; a or b or 10 11 The equation of the line through the three collinear points 12 13 The equation of the plane through the four coplanar points Exercise Set 10.2 , ; maximum value of No feasible solutions Unbounded solution Invest $6000 in bond A and $4000 in bond B; the annual yield is $880 cup of milk, a ounces of corn flakes; minimum and b c are nonbinding; for for is binding is binding and for is nonbinding and for yields the empty set yields the empty set 550 containers from company A and 300 containers from company B; maximum shipping 925 containers from company A and no containers from company B; maximum shipping 0.4 pound of ingredient A and 2.4 pounds of ingredient B; minimum Exercise Set 10.3 700 a b 4 a Ox, units; sheep, b First kind, measure; second kind, a measure; third kind, , b Exercise 7(b); gold, unit measure , minae; brass, minae; tin, minae; iron, a where t is an arbitrary number b Take , so that , , , c Take , so that , , , minae a Legitimate son, b Gold, staters; illegitimate son, minae; brass, minae; tin, c First person, 45; second person, a b a The cubic runout spline b Maximum at Maximum at a b c The three data points are collinear (b) (b) Exercise Set 10.5 a b P is regular since all entries of P are positive; a minae; iron, ; third person, Exercise Set 10.4 staters minae b P is regular, since all entries of P are positive: a b c a Thus, no integer power of P has all positive entries b c as n increases, so The entries of the limiting vector for any are not all positive has all positive entries; in region 1, Exercise Set 10.6 a b c a in region 2, and in region as n increases b c a b c (a) (c) The th entry is the number of family members who influence both the ith and jth family members a b c and a None b First, A; second, B and E (tie); fourth, C; fifth, D Exercise Set 10.7 a b c Let , for example a b c d a b c d e Exercise Set 10.8 a b c a Use Corollary 10.8.4; all row sums are less than one b Use Corollary 10.8.5; all column sums are less than one c Use Theorem 10.8.3, with has all positive entries Price of tomatoes, $120.00; price of corn, $100.00; price of lettuce, $106.67 $1256 for the CE, $1448 for the EE, $1556 for the ME (b) Exercise Set 10.9 The second class; $15,000 $223 Exercise Set 10.10 a b c d (b) (c) a b c a b a b a b Exercise Set 10.11 a b c d for and , Exercise Set 10.12 (c) a b Same as part (a) c %; for and , % , , Exercise Set 10.13 ; Rotation angles: (upper left); (upper right); (lower left); (lower right); a (i) ; (ii) all rotation angles are b (i) ; (ii) all rotation angles are c (i) ; (ii) rotation angles: (top); d (i) ; (ii) rotation angles: (upper left); ; (iii) This set is a fractal ; (iii) This set is a fractal (lower left); (upper right); (0.766, 0.996) rounded to three decimal places 10 ; the cube is not a fractal ; ; ; the set is a fractal (lower right); (iii) (lower right) (iii) This set is a fractal This set is a fractal 11 12 Area of ; area of ; area of ; area of ; area of Exercise Set 10.14 , , One 1-cycle: , , , ; one 3-cycle: , , ; two 4-cycles: and two 12-cycles: ; and , , (a) 3, 7, 10, 2, 12, 14, 11, 10, 6, 1, 7, 8, 0, 8, 8, 1, 9, 10, 4, 14, 3, 2, 5, 7, 12, 4, 1, 5, 6, 11, 2, 13, 0, 13, 13, 11, 9, 5, 14, 4, 3, 7, (c) (5, 5), (10, 15), (4, 19), (2, 0), (2, 2), (4, 6), (10, 16), (5, 0), (5, 5), (c) The first five iterates of (b) are , The matrices of Anosov automorphisms are , and (c) The transformation affects a rotation of S through , , and in the clockwise direction In region I: 12 and ; in region II: form one 2-cycle, and ; in region III: and ; in region IV: form another 2-cycle 14 Begin with a array of white pixels and add the letter ‘A’ in black pixels to it Apply the mapping to this image, which will scatter the black pixels throughout the image Then superimpose the letter ‘B’ in black pixels onto this image Apply the mapping again and then superimpose the letter ‘C’ in black pixels onto the resulting image Repeat this procedure with the letters ‘D’ and ‘E’ The next application of the mapping will return you to the letter ‘A’ with the pixels for the letters ‘B’ through ‘E’ scattered in the background Exercise Set 10.15 a GIYUOKEVBH b SFANEFZWJH a b Not invertible c d Not invertible e Not invertible f WE LOVE MATH THEY SPLIT THE ATOM I HAVE COME TO BURY CAESAR a 010110001 b A is invertible modulo 29 if and only if (mod 29) Exercise Set 10.16 Eigenvalues: , ; eigenvectors: 12 generations; 006% ; Exercise Set 10.17 a b c 2.375 1.49611 Exercise Set 10.18 a of population; b of population; ; harvest 57.9% of youngest age class Exercise Set 10.19 Exercise Set 10.20 a Yes; b No; c Yes; d Yes; number of triangles a number of vertex points , number of boundary vertex points ; Equation (7) is b a b c d a Two of the coefficients are zero b At least one of the coefficients is zero c None of the coefficients are zero a b Copyright © 2010 John Wiley & Sons, Inc All rights reserved ... This edition of Elementary Linear Algebra gives an introductory treatment of linear algebra that is suitable for a first undergraduate course Its aim is to present the fundamentals of linear algebra. .. Chris Rorres entitled Elementary Linear Algebra: Applications Version, 10th (ISBN 9780470432051), whose purpose is to supplement this version with an extensive body of applications However, to accommodate... functions The following are linear equations: The following are not linear equations: A finite set of linear equations is called a system of linear equations or, more briefly, a linear system The variables