Linear Algebra and Its Applications

Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independenEditor-in-Chief: Deirdre lynchSenior Acquisitions Editor: William HofmannSponsoring Editor: Caroline celanoSenior content editor: Chere bemelmansEditorial Assistant: Brandon rawnsleySenior Managing Editor: Karen WernholmAssociate Managing Editor: Tamela AmbushDigital Assets Manager: Marianne grothSupplements Production Coordinator: Kerri McQueerSenior media producer: Carl CottrellQA Manager, Assessment Content: Marty WrighExecutive Marketing Manager Jeff WeidenaarMarketing assistant Kendra bassiSenior Author Support/Technology Specialist: Joe vetereRights and permissions advisor: Michael JoyceImage manager. Rachel youdelmanSenior Manufacturing Buyer: Carol melvilleScnior Media buyer: Ginny MichaudDesign manager: Andrea nixSenior designer: Beth paquinText DeAndrea nixProduction Coordination Tamela ambushComposition: Dennis KletzingIllustrations: Scientific illustratorsCover Design: Nancy Goulet, StudiowinkCover Image: Shoula/Stone/Getty ImagesFor permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on page Pl, which ishereby made part of this copyright pageMany of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Wherethose designations appear in this book, and Pearson Education was aware of a trademark claim, the designations have beenprinted in initial caps or all capsLibrary of Congress Cataloging-in-Publication DataLay, David CLinear algebra and its applications /David C. Lay. -4th ed. updateP. CI.Includes indexISBN-13:978-0-321-38517-8ISBN-10:0-321-38517-91. Algebras. Linear-Textbooks. [. TitleQA184.2.L392012512.5dc222010048460Copyright C)2012, 2006, 1997,1994 Pearson Education, IncAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any formor by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of thepublisher. Printed in the United States of America. For information on obtaining permission for use of material in this work,please submit a written request to Pearson Education, Inc, Rights and Contracts Department, 501 Boylston Street, Suite900,Boston,Ma02116,faxyourrequestto617-671-3447,ore-mailathttp://www.pearsoned.com/legal/permissions.htm12345678910—DOW-1413121110Addison-Wesleyis an imprint ofPEARSONISBN13:978-0-321-38517-8www.pearsonhighered.comISBN10:0-321-385179To my wife, Lillian, and our children,Christina Deborah and melissa, whosesupport, encouragement, and faithfulprayers made this book possibleAbout the authorDavid C. Lay holds a B.A. from Aurora University (Illinois), and an M.A. and Ph. Dfrom the University of california at Los angeles. lay has been an educator and researchmathenatician since 1966, mostly at the University of Mary land, College Park. He hasalso served as a visiting professor at the University of Amsterdam, the Free Universityin Amsterdam, and the University of Kaiserslautern, Germany. He has published morethan 30 research articles on functional analysis and linear algebraAs a founding member of the NsF-sponsored Linear Algebra Curriculum StudyGroup, Lay has been a leader in the current movement to modernize the linear algebracurriculum. Lay is also a co-author of several mathematics texts, including Introduction to Functional Analysis with Angus E. Taylor, Calculus and Its Applications, withL J. Goldstein and D. I Schneider, and Linear Algebra Gems-Assets for Undergrad-uate Mathematics, with D. Carlson. C.R. Johnson. and A. D. Porter.Professor Lay has received four university awards for teaching excellence, includ-ing, in 1996, the title of Distinguished Scholar-Teacher of the University of MarylandIn 1994, he was given one of the Mathematical Association of America's Awards forDistinguished College or University Teaching of Mathematics. He has been electedby the university students to membership in Alpha Lambda Delta National ScholasticHonor Society and golden Key National Honor Society. In 1989, Aurora Universitconferred on him the Outstanding Alumnus award. Lay is a member of the AmericanMathenatical society, the Canadian Mathenatical Society, the International LinearAlgebra Society, the Mathematical Association of America, Sigma Xi, and the Societfor Industrial and Applied Mathematics. Since 1992, he has served several terms on thenational board of the association of christians in the mathematical SciencesContentsPrefaceA Note to students xyChapter 1 Linear Equations in Linear Algebra 1INTRODUCTORY EXAMPLE: Linear Models in Economics and EngineeringSystems of Linear equations 21.2 Row Reduction and echelon forms 12Ⅴ ector equations241.4The Matrix Equation Ax =b 341.5 Solution Sets of Linear Systems 436 Applications of Linear Systems 491.7Linear Independence 55Introduction to linear transformations 62The matrix of a linear transformation 701.10 Linear Models in Business, Science, and Engineering 80Supplementary Exercises 88Chapter 2 Matrix Algebra 91INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design 91Matrix Operations 922.2 The Inverse of a matrix 1023 Characterizations of invertible matrices ll12.4Partitioned matrices 117Matrix Factorizations 1236 The Leontief Input-Output Model 1322.7 Applications to Computer Graphics 1382.8 Subspaces of r" 1462.9Dimension and rank 153Supplementary Exercises 160Chapter 3 Determinants 163INTRODUCTORY EXAMPLE, Random paths and distortion 163Introduction to determinants 1643.2 Properties of Determinants 169vi Contents3 Cramer's Rule Volume and Linear Trans formations 177Supplementary Exercises 185Chapter 4 Vector Spaces 189INTRODUCTORY EXAMPLE: Space Flight and Control Systems 1894.1Vector Spaces and Subspaces 1904.2 Null Spaces, Column Spaces, and Linear Transformations 1984.3Linearly Independent Sets: Bases 2084.4Coordinate Systems 2164.5The Dimension of a vector Space 2254.6Rank 2304.7Change of basis 2394.8 Applications to Difference Equations 2444.9Applications to markov chains 253Supplementary Exercises 262Chapter 5 Eigenvalues and Eigenvectors 265INTRODUCTORY EXAMPLE: Dynamical Systcms and Spotted Owls 265Eigenvectors and eigenvalues 2665.2The Characteristic Equation 2735.3 Diagonalization 2815.4 Eigenvectors and Linear Transformations 2885Complex Eigenvalues 2955.6Discrete Dynamical Systems 3015.7 Applications to Differential Equations 3115.8Iterative Estimates for Eigenvalues 319Supplementary Exercises 326Chapter 6 Orthogonality and Least Squares 329INTRODUCTORY EXAMPLE: The north American Datumand GPS Navigation 3296.1Inner Product Length, and orthlity 330Orthogonal Sets 3386.3al projectio3476.4The gram-Schmidt plLeast-Squares Problems 3606.6pplicationsModels 3686.7Inner Product Spaces 3766.8Applications of Inner Product Spaces 3Supplementary Exercises 390Contents viiChapter 7 Symmetric Matrices and Quadratic Forms 393INTRODUCTORY EXAMPLE: Multichannel Image Processing 3937.1Diagonaliz ation of symmetric matrices 3957.2 Quadratic Fo7.3 Constrained Optimization 40874The singular value Decomposition 4147.5 Applications to Image Processing and Statistics 424Supplementary exercises 432Chapter 8 The Geometry of Vector Spaces 435INTRODUCTORY EXAMPLE: The platonic solids 4358.1 Affine combinations 4368.2 Affine Independence 4448.3Convex Com binations 4548.4hyperplanes 4618.5Polite4698.6Curves and Surfaces 481Chapter 9 Optimization (Online)INTRODUCTORY EXAMPLE: The Berlin Airlift9.1Matrix games9.2 Linear programming -Geometric method9.3Linear Programming-Simplex Method9.4DualitChapter 10 Finite-State Markov Chains(Online)NTRODUCTORY EXAMPLE: Google and Markov chains10.1 Introduction and Examples10.2 The Steady-State Vector and googles PageRank10.3 Communication Classes10.4 Classification of States and Periodicity10.5 The Fundamental matrix10.6 Markov Chains and Baseball Statisticsviii ContentsAppendixesAUniqueness of the Reduced Echelon Form A1BComplex Numbers A2Glossary A7Answers to odd-Numbered Exercises A17Index 1Photo credits PlPrefaceThe response of students and teachers to the first three editions of Linear algebra andIts Applications has been most gratifying. This Fourth Edition provides substantialsupport both for teaching and for using technology in the course. As before, the textprovides a modern elementary introduction to linear algebra and a broad selection ofinteresting applications. The material is accessible to students with the maturity thatshould come from successful completion of two semesters of college-level mathematicsusually calculusThe main goal of the text is to help students master the basic concepts and skills theywill use later in their careers. The topics here follow the recommendations of the linearAlgebra CurriculuM Study Group, which were based on a careful investigation of thereal needs of the students and a consensus among professionals in many disciplines thatuse linear algebra. Hopefully, this course will be one of the most useful and interestingmathematics classes taken by undergraduatesWHATS NEW IN THIS EDITIONThe main goal of this revision was to update the exercises and provide additional content both in the book and online1. More than 25 percent of the exercises are new or updated, especially the computational exercises. The exercise sets remain one of the most important features of thisbook, and these new exercises follow the same high standard of the exercise sets ofthe past three editions. They are crafted in a way that retells the substance of eachof the sections they follow, developing the students'confidence while challengingthem to practice and generalize the new ideas they have just encountered2. Twenty-five percent of chapter openers are new. These introductory vignettes provide applications of linear algebra and the motivation for developing the mathematicsthat follows. The text returns to that application in a section toward the end of thechapter3. A New Chapter: Chapter 8, The Geometry of Vector Spaces, provides a fresh topicthat my students have really enjoyed studying. Sections 1, 2, and 3 provide the basicgeometric tools. Then Section 6 uses these ideas to study Bezier curves and surfaceswhich are used in engineering and online computer graphics (in Adobe Illustratorand Macromedia FreeHand). These four sections can be covered in four or five50-minute clA second course in linear algebra applications typically begins with a substantialreview of key ideas from the first course. If part of Chapter 8 is in the first course,the second course could include a brief review of sections 1 to 3 and then a focus onthe geometry in sections 4 and 5. That would lead naturally into the online chapters9 and 10, which have been used with Chapter at a number of schools for the pastfive4. The Study Guide, which has always been an integral part of the book, has been udated to cover the new Chapter 8. As with past editions, the Study Guide incorporates