Linear Algebra and Its Applications (2021) PDF

This is a special edition of an established title widely used by colleges and GLOBAL universities throughout the world. Pearson published this exclusive edition for the benefit of students outside the...

This is a special edition of an established title widely used by colleges and GLOBAL universities throughout the world. Pearson published this exclusive edition for the benefit of students outside the United States and Canada. If you GLOBAL EDITION purchased this book within the United States or Canada, you should be aware EDITION EDITION GLOB AL that it has been imported without the approval of the Publisher or Author. Linear Algebra and Its Applications Linear Algebra and Its Applications, now in its sixth edition, not only follows the recommendations of the original Linear Algebra Curriculum Study Group (LACSG) but also includes ideas currently being discussed by the LACSG 2.0 and continues to provide a modern elementary introduction to linear algebra. This edition adds exciting new topics, examples, and online resources to highlight the linear algebraic foundations of machine learning, artificial intelligence, data science, and digital signal processing. Features Many fundamental ideas of linear algebra are introduced early, in the concrete setting of n , and then gradually examined from different points of view. Utilizing a modern view of matrix multiplication simplifies many arguments and ties vector space ideas into the study of linear systems. Every major concept is given a geometric interpretation to help students learn better by visualizing the idea. Keeping with the recommendations of the original LACSG, because orthogonality plays an important role in computer calculations and numerical linear algebra, and because inconsistent linear systems arise so often in practical work, this title includes a comprehensive treatment of both orthogonality and the least-squares problem. Projects at the end of each chapter on a wide range of themes (including using linear transformations to create art and detecting and correcting errors in encoded messages) enhance EDITION student learning. SIXTH NEW! Reasonable Answers advice and exercises encourage students to ensure their computations are consistent with the data at hand and the questions being asked. Available separately for purchase is MyLab Math for Linear Algebra and Its Applications, the teaching Linear Algebra and Its Applications Lay Lay McDonald and learning platform that empowers instructors to personalize learning for every student. When combined with Pearson’s trusted educational content, this optional suite helps deliver the learning outcomes desired. This edition includes interactive versions of many of the figures in the SIXTH EDITION text, letting students manipulate figures and experiment with matrices to gain a deeper geometric understanding of key concepts and principles. David C. Lay Steven R. Lay Judi J. McDonald CVR_LAY1216_06_GE_CVR_Vivar.indd 1 09/04/21 12:22 PM S I X T H E D I T I O N Linear Algebra and Its Applications G L O B A L E D I T I O N David C. Lay University of Maryland–College Park Steven R. Lay Lee University Judi J. McDonald Washington State University Pearson Education Limited KAO Two KAO Park Hockham Way Harlow Essex CM17 9SR United Kingdom and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited, 2022 The rights of David C. Lay, Steven R. Lay, and Judi J. McDonald to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Authorized adaptation from the United States edition, entitled Linear Algebra and Its Applications, 6th Edition, ISBN 978-0-13-585125-8 by David C. Lay, Steven R. Lay, and Judi J. McDonald, published by Pearson Education © 2021. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. For information regarding permissions, request forms, and the appropriate contacts within the Pearson Education Global Rights and Permissions department, please visit www.pearsoned.com/permissions. This eBook is a standalone product and may or may not include all assets that were part of the print version. It also does not provide access to other Pearson digital products like MyLab and Mastering. The publisher reserves the right to remove any material in this eBook at any time. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 10: 1-292-35121-7 ISBN 13: 978-1-292-35121-6 eBook ISBN 13: 978-1-292-35122-3  To my wife, Lillian, and our children, Christina, Deborah, and Melissa, whose support, encouragement, and faithful prayers made this book possible. David C. Lay About the Authors David C. Lay As a founding member of the NSF-sponsored Linear Algebra Curriculum Study Group (LACSG), David Lay was a leader in the movement to modernize the linear algebra curriculum and shared those ideas with students and faculty through his authorship of the first four editions of this textbook. David C. Lay earned a B.A. from Aurora University (Illinois), and an M.A. and Ph.D. from the University of California at Los Angeles. David Lay was an educator and research mathematician for more than 40 years, mostly at the University of Maryland, College Park. He also served as a visiting professor at the University of Amsterdam, the Free University in Amsterdam, and the University of Kaiserslautern, Germany. He published more than 30 research articles on functional analysis and linear algebra. Lay was also a coauthor of several mathematics texts, including Introduction to Functional Analysis with Angus E. Taylor, Calculus and Its Applications, with L. J. Goldstein and D. I. Schneider, and Linear Algebra Gems—Assets for Undergraduate Mathematics, with D. Carlson, C. R. Johnson, and A. D. Porter. David Lay received four university awards for teaching excellence, including, in 1996, the title of Distinguished Scholar-Teacher of the University of Maryland. In 1994, he was given one of the Mathematical Association of America’s Awards for Distin- guished College or University Teaching of Mathematics. He was elected by the univer- sity students to membership in Alpha Lambda Delta National Scholastic Honor Society and Golden Key National Honor Society. In 1989, Aurora University conferred on him the Outstanding Alumnus award. David Lay was a member of the American Mathe- matical Society, the Canadian Mathematical Society, the International Linear Algebra Society, the Mathematical Association of America, Sigma Xi, and the Society for In- dustrial and Applied Mathematics. He also served several terms on the national board of the Association of Christians in the Mathematical Sciences. In October 2018, David Lay passed away, but his legacy continues to benefit students of linear algebra as they study the subject in this widely acclaimed text. 3 4 About the Authors Steven R. Lay Steven R. Lay began his teaching career at Aurora University (Illinois) in 1971, after earning an M.A. and a Ph.D. in mathematics from the University of California at Los Angeles. His career in mathematics was interrupted for eight years while serving as a missionary in Japan. Upon his return to the States in 1998, he joined the mathematics faculty at Lee University (Tennessee) and has been there ever since. Since then he has supported his brother David in refining and expanding the scope of this popular linear algebra text, including writing most of Chapters 8 and 9. Steven is also the author of three college-level mathematics texts: Convex Sets and Their Applications, Analysis with an Introduction to Proof, and Principles of Algebra. In 1985, Steven received the Excellence in Teaching Award at Aurora University. He and David, and their father, Dr. L. Clark Lay, are all distinguished mathematicians, and in 1989, they jointly received the Outstanding Alumnus award from their alma mater, Aurora University. In 2006, Steven was honored to receive the Excellence in Scholarship Award at Lee University. He is a member of the American Mathematical Society, the Mathematics Association of America, and the Association of Christians in the Mathematical Sciences. Judi J. McDonald Judi J. McDonald became a co-author on the fifth edition, having worked closely with David on the fourth edition. She holds a B.Sc. in Mathematics from the University of Alberta, and an M.A. and Ph.D. from the University of Wisconsin. As a professor of Mathematics, she has more than 40 publications in linear algebra research journals and more than 20 students have completed graduate degrees in linear algebra under her supervision. She is an associate dean of the Graduate School at Washington State Uni- versity and a former chair of the Faculty Senate. She has worked with the mathematics outreach project Math Central (http://mathcentral.uregina.ca/) and is a member of the second Linear Algebra Curriculum Study Group (LACSG 2.0). Judi has received three teaching awards: two Inspiring Teaching awards at the Uni- versity of Regina, and the Thomas Lutz College of Arts and Sciences Teaching Award at Washington State University. She also received the College of Arts and Sciences Insti- tutional Service Award at Washington State University. Throughout her career, she has been an active member of the International Linear Algebra Society and the Association for Women in Mathematics. She has also been a member of the Canadian Mathematical Society, the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. Contents About the Authors 3 Preface 12 A Note to Students 22 Chapter 1 Linear Equations in Linear Algebra 25 INTRODUCTORY EXAMPLE: Linear Models in Economics and Engineering 25 1.1 Systems of Linear Equations 26 1.2 Row Reduction and Echelon Forms 37 1.3 Vector Equations 50 1.4 The Matrix Equation Ax D b 61 1.5 Solution Sets of Linear Systems 69 1.6 Applications of Linear Systems 77 1.7 Linear Independence 84 1.8 Introduction to Linear Transformations 91 1.9 The Matrix of a Linear Transformation 99 1.10 Linear Models in Business, Science, and Engineering 109 Projects 117 Supplementary Exercises 117 Chapter 2 Matrix Algebra 121 INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design 121 2.1 Matrix Operations 122 2.2 The Inverse of a Matrix 135 2.3 Characterizations of Invertible Matrices 145 2.4 Partitioned Matrices 150 2.5 Matrix Factorizations 156 2.6 The Leontief Input–Output Model 165 2.7 Applications to Computer Graphics 171 5 6 Contents 2.8 Subspaces of Rn 179 2.9 Dimension and Rank 186 Projects 193 Supplementary Exercises 193 Chapter 3 Determinants 195 INTRODUCTORY EXAMPLE: Weighing Diamonds 195 3.1 Introduction to Determinants 196 3.2 Properties of Determinants 203 3.3 Cramer’s Rule, Volume, and Linear Transformations 212 Projects 221 Supplementary Exercises 221 Chapter 4 Vector Spaces 225 INTRODUCTORY EXAMPLE: Discrete-Time Signals and Digital Signal Processing 225 4.1 Vector Spaces and Subspaces 226 4.2 Null Spaces, Column Spaces, Row Spaces, and Linear Transformations 235 4.3 Linearly Independent Sets; Bases 246 4.4 Coordinate Systems 255 4.5 The Dimension of a Vector Space 265 4.6 Change of Basis 273 4.7 Digital Signal Processing 279 4.8 Applications to Difference Equations 286 Projects 295 Supplementary Exercises 295 Chapter 5 Eigenvalues and Eigenvectors 297 INTRODUCTORY EXAMPLE: Dynamical Systems and Spotted Owls 297 5.1 Eigenvectors and Eigenvalues 298 5.2 The Characteristic Equation 306 5.3 Diagonalization 314 5.4 Eigenvectors and Linear Transformations 321 5.5 Complex Eigenvalues 328 5.6 Discrete Dynamical Systems 335 5.7 Applications to Differential Equations 345 5.8 Iterative Estimates for Eigenvalues 353 5.9 Applications to Markov Chains 359 Projects 369 Supplementary Exercises 369  Contents 7 Chapter 6 Orthogonality and Least Squares 373 INTRODUCTORY EXAMPLE: Artificial Intelligence and Machine Learning 373 6.1 Inner Product, Length, and Orthogonality 374 6.2 Orthogonal Sets 382 6.3 Orthogonal Projections 391 6.4 The Gram–Schmidt Process 400 6.5 Least-Squares Problems 406 6.6 Machine Learning and Linear Models 414 6.7 Inner Product Spaces 423 6.8 Applications of Inner Product Spaces 431 Projects 437 Supplementary Exercises 438 Chapter 7 Symmetric Matrices and Quadratic Forms 441 INTRODUCTORY EXAMPLE: Multichannel Image Processing 441 7.1 Diagonalization of Symmetric Matrices 443 7.2 Quadratic Forms 449 7.3 Constrained Optimization 456 7.4 The Singular Value Decomposition 463 7.5 Applications to Image Processing and Statistics 473 Projects 481 Supplementary Exercises 481 Chapter 8 The Geometry of Vector Spaces 483 INTRODUCTORY EXAMPLE: The Platonic Solids 483 8.1 Affine Combinations 484 8.2 Affine Independence 493 8.3 Convex Combinations 503 8.4 Hyperplanes 510 8.5 Polytopes 519 8.6 Curves and Surfaces 531 Project 542 Supplementary Exercises 543 Chapter 9 Optimization 545 INTRODUCTORY EXAMPLE: The Berlin Airlift 545 9.1 Matrix Games 546 9.2 Linear Programming Geometric Method 560 9.3 Linear Programming Simplex Method 570 9.4 Duality 585 Project 594 Supplementary Exercises 594 8 Contents Chapter 10 Finite-State Markov Chains C-1 (Available Online) INTRODUCTORY EXAMPLE: Googling Markov Chains C-1 10.1 Introduction and Examples C-2 10.2 The Steady-State Vector and Google’s PageRank C-13 10.3 Communication Classes C-25 10.4 Classification of States and Periodicity C-33 10.5 The Fundamental Matrix C-42 10.6 Markov Chains and Baseball Statistics C-54 Appendixes A Uniqueness of the Reduced Echelon Form 597 B Complex Numbers 599 Credits 604 Glossary 605 Answers to Odd-Numbered Exercises A-1 Index I-1 Applications Index Biology and Ecology Color monitors, 178 Series and shunt circuits, 161 Estimating systolic blood pressure, 422 Computer graphics, 122, 171–177, Transfer matrix, 161–162, 163 Laboratory animal trials, 367 498–500 Molecular modeling, 173–174 Cray supercomputer, 153 Engineering Net primary production of nutrients, Data storage, 66, 163 Aircraft performance, 422, 437 418–419 Error-detecting and error-correcting Boeing Blended Wing Body, 122 Nutrition problems, 109–111, 115 codes, 447, 471 Cantilevered beam, 293 Predator-prey system, 336–337, 343 Game theory, 519 CFD and aircraft design, 121–122 Spotted owls and stage-matrix models, High-end computer graphics boards, 176 Deflection of an elastic beam, 137, 144 297–298, 341–343 Homogeneous coordinates, 172–173, 174 Deformation of a material, 482 Business and Economics Parallel processing, 25, 132 Equilibrium temperatures, 36, 116–117, Accelerator-multiplier model, 293 Perspective projections, 175–176 193 Average cost curve, 418–419 Vector pipeline architecture, 153 Feedback controls, 519 Car rental fleet, 116, 368 Virtual reality, 174 Flexibility and stiffness matrices, 137, Cost vectors, 57 VLSI microchips, 150 144 Equilibrium prices, 77–79, 82 Wire-frame models, 121, 171 Heat conduction, 164 Exchange table, 82 Image processing, 441–442, 473–474, Feasible set, 460, 562 Control Theory 479 Gross domestic product, 170 Controllable system, 296 LU factorization and airflow, 122 Indifference curves, 460–461 Control systems engineering, 155 Moving average filter, 293 Intermediate demand, 165 Decoupled system, 340, 346, 349 Superposition principle, 95, 98, 112 Investment, 294 Deep space probe, 155 Leontief exchange model, 25, 77–79 State-space model, 296, 335 Mathematics Leontief input–output model, 25, Steady-state response, 335 Area and volume, 195–196, 215–217 165–171 Transfer function (matrix), 155 Attractors/repellers in a dynamical Linear programming, 26, 111–112, 153, system, 338, 341, 343, 347, 351 484, 519, 522, 560–566 Electrical Engineering Bessel’s inequality, 438 Loan amortization schedule, 293 Branch and loop currents, 111–112 Best approximation in function spaces, Manufacturing operations, 57, 96 Circuit design, 26, 160 426–427 Marginal propensity to consume, 293 Current flow in networks, 111–112, Cauchy-Schwarz inequality, 427 Markov chains, 311, 359–368, C-1–C-63 115–116 Conic sections and quadratic surfaces, Maximizing utility subject to a budget Discrete-time signals, 228, 279–280 481 constraint, 460–461 Inductance-capacitance circuit, 242 Differential equations, 242, 345–347 Population movement, 113, 115–116, Kirchhoff’s laws, 161 Fourier series, 434–436 311, 361 Ladder network, 161, 163–164 Hermite polynomials, 272 Price equation, 170 Laplace transforms, 155, 213 Hypercube, 527–529 Total cost curve, 419 Linear filters, 287–288 Interpolating polynomials, 49, 194 Value added vector, 170 Low-pass filter, 289, 413 Isomorphism, 188, 260–261 Variable cost model, 421 Minimal realization, 162 Jacobian matrix, 338 Computers and Computer Science Ohm’s law, 111–113, 161 Laguerre polynomials, 272 Bézier curves and surfaces, 509, 531–532 RC circuit, 346–347 Laplace transforms, 155, 213 CAD, 537, 541 RLC circuit, 254 Legendre polynomials, 430 Page numbers denoted with “C” are found within the online chapter 10 Linear transformations in calculus, 241, QR algorithm, 312–313, 357 Radar data, 155 324–325 QR factorization, 403–404, 405, 413, 438 Seismic data, 25 Simplex, 525–527 Rank-revealing factorization, 163, 296, Space probe, 155 Splines, 531–534, 540–541 481 Steady-state heat flow, 36, 164 Triangle inequality, 427 Rayleigh quotient, 358, 439 Superposition principle, 95, 98, 112 Trigonometric polynomials, 434 Relative error, 439 Three-moment equation, 293 Schur complement, 154 Traffic flow, 80 Numerical Linear Algebra Schur factorization, 439 Trend surface, 419 Band matrix, 164 Singular value decomposition, 163, Weather, 367 Block diagonal matrix, 153, 334 463–473 Wind tunnel experiment, 49 Cholesky factorization, 454–455, 481 Sparse matrix, 121, 168, 206 Companion matrix, 371 Spectral decomposition, 446–447 Condition number, 147, 149, 211, 439, Spectral factorization, 163 Statistics 469 Tridiagonal matrix, 164 Analysis of variance, 408, 422 Effective rank, 190, 271, 465 Vandermonde matrix, 194, 371 Covariance, 474–476, 477, 478, 479 Floating point arithmetic, 33, 45, 221 Vector pipeline architecture, 153 Full rank, 465 Fundamental subspaces, 379, 439, Least-squares error, 409 469–470 Physical Sciences Least-squares line, 413, 414–416 Givens rotation, 119 Cantilevered beam, 293 Linear model in statistics, 414–420 Gram matrix, 482 Center of gravity, 60 Markov chains, 359–360 Gram–Schmidt process, 405 Chemical reactions, 79, 83 Mean-deviation form for data, 417, 475 Hilbert matrix, 149 Crystal lattice, 257, 263 Moore-Penrose inverse, 471 Householder reflection, 194 Decomposing a force, 386 Multichannel image processing, Ill-conditioned matrix (problem), 147 Gaussian elimination, 37 441–442, 473–479 Inverse power method, 356–357 Hooke’s law, 137 Multiple regression, 419–420 Iterative methods, 353–359 Interpolating polynomial, 49, 194 Orthogonal polynomials, 427 Jacobi’s method for eigenvalues, 312 Kepler’s first law, 422 Orthogonal regression, 480–481 LAPACK, 132, 153 Landsat image, 441–442 Powers of a matrix, 129 Large-scale problems, 119, 153 Linear models in geology and geography, LU factorization, 157–158, 162–163, 164 419–420 Principal component analysis, 441–442, Operation counts, 142, 158, 160, 206 Mass estimates for radioactive 476–477 Outer products, 133, 152 substances, 421 Quadratic forms in statistics, 449 Parallel processing, 25 Mass-spring system, 233, 254 Regression coefficients, 415 Partial pivoting, 42, 163 Model for glacial cirques, 419 Sums of squares (in regression), 422, Polar decomposition, 482 Model for soil pH, 419 431–432 Power method, 353–356 Pauli spin matrices, 194 Trend analysis, 433–434 Powers of a matrix, 129 Periodic motion, 328 Variance, 422, 475–476 Pseudoinverse, 470, 482 Quadratic forms in physics, 449–454 Weighted least-squares, 424, 431–432 This page is intentionally left blank Preface The response of students and teachers to the first five editions of Linear Algebra and Its Applications has been most gratifying. This Sixth Edition provides substantial support both for teaching and for using technology in the course. As before, the text provides a modern elementary introduction to linear algebra and a broad selection of interesting classical and leading-edge applications. The material is accessible to students with the maturity that should come from successful completion of two semesters of college-level mathematics, usually calculus. The main goal of the text is to help students master the basic concepts and skills they will use later in their careers. The topics here follow the recommendations of the original Linear Algebra Curriculum Study Group (LACSG), which were based on a careful investigation of the real needs of the students and a consensus among professionals in many disciplines that use linear algebra. Ideas being discussed by the second Linear Algebra Curriculum Study Group (LACSG 2.0) have also been included. We hope this course will be one of the most useful and interesting mathematics classes taken by undergraduates. What’s New in This Edition The Sixth Edition has exciting new material, examples, and online resources. After talk- ing with high-tech industry researchers and colleagues in applied areas, we added new topics, vignettes, and applications with the intention of highlighting for students and faculty the linear algebraic foundational material for machine learning, artificial intelli- gence, data science, and digital signal processing. Content Changes Since matrix multiplication is a highly useful skill, we added new examples in Chap- ter 2 to show how matrix multiplication is used to identify patterns and scrub data. Corresponding exercises have been created to allow students to explore using matrix multiplication in various ways. In our conversations with colleagues in industry and electrical engineering, we heard repeatedly how important understanding abstract vector spaces is to their work. After reading the reviewers’ comments for Chapter 4, we reorganized the chapter, condens- ing some of the material on column, row, and null spaces; moving Markov chains to the end of Chapter 5; and creating a new section on signal processing. We view signals 12  Preface 13 as an infinite dimensional vector space and illustrate the usefulness of linear trans- formations to filter out unwanted “vectors” (a.k.a. noise), analyze data, and enhance signals. By moving Markov chains to the end of Chapter 5, we can now discuss the steady state vector as an eigenvector. We also reorganized some of the summary material on determinants and change of basis to be more specific to the way they are used in this chapter. In Chapter 6, we present pattern recognition as an application of orthogonality, and the section on linear models now illustrates how machine learning relates to curve fitting. Chapter 9 on optimization was previously available only as an online file. It has now been moved into the regular textbook where it is more readily available to faculty and students. After an opening section on finding optimal strategies to two-person zero- sum games, the rest of the chapter presents an introduction to linear programming— from two-dimensional problems that can be solved geometrically to higher dimen- sional problems that are solved using the Simplex Method. Other Changes In the high-tech industry, where most computations are done on computers, judging the validity of information and computations is an important step in preparing and analyzing data. In this edition, students are encouraged to learn to analyze their own computations to see if they are consistent with the data at hand and the questions being asked. For this reason, we have added “Reasonable Answers” advice and exercises to guide students. We have added a list of projects to the end of each chapter (available online and in MyLab Math). Some of these projects were previously available online and have a wide range of themes from using linear transformations to create art to exploring additional ideas in mathematics. They can be used for group work or to enhance the learning of individual students. PowerPoint lecture slides have been updated to cover all sections of the text and cover them more thoroughly. Distinctive Features Early Introduction of Key Concepts Many fundamental ideas of linear algebra are introduced within the first seven lectures, in the concrete setting of Rn , and then gradually examined from different points of view. Later generalizations of these concepts appear as natural extensions of familiar ideas, visualized through the geometric intuition developed in Chapter 1. A major achievement of this text is that the level of difficulty is fairly even throughout the course. A Modern View of Matrix Multiplication Good notation is crucial, and the text reflects the way scientists and engineers actually use linear algebra in practice. The definitions and proofs focus on the columns of a matrix rather than on the matrix entries. A central theme is to view a matrix–vector product Ax as a linear combination of the columns of A. This modern approach simplifies many arguments, and it ties vector space ideas into the study of linear systems. 14 Preface Linear Transformations Linear transformations form a “thread” that is woven into the fabric of the text. Their use enhances the geometric flavor of the text. In Chapter 1, for instance, linear transfor- mations provide a dynamic and graphical view of matrix–vector multiplication. Eigenvalues and Dynamical Systems Eigenvalues appear fairly early in the text, in Chapters 5 and 7. Because this material is spread over several weeks, students have more time than usual to absorb and review these critical concepts. Eigenvalues are motivated by and applied to discrete and continuous dynamical systems, which appear in Sections 1.10, 4.8, and 5.9, and in five sections of Chapter 5. Some courses reach Chapter 5 after about five weeks by covering Sections 2.8 and 2.9 instead of Chapter 4. These two optional sections present all the vector space concepts from Chapter 4 needed for Chapter 5. Orthogonality and Least-Squares Problems These topics receive a more comprehensive treatment than is commonly found in be- ginning texts. The original Linear Algebra Curriculum Study Group has emphasized the need for a substantial unit on orthogonality and least-squares problems, because orthogonality plays such an important role in computer calculations and numerical linear algebra and because inconsistent linear systems arise so often in practical work. Pedagogical Features Applications A broad selection of applications illustrates the power of linear algebra to explain fundamental principles and simplify calculations in engineering, computer science, mathematics, physics, biology, economics, and statistics. Some applications appear in separate sections; others are treated in examples and exercises. In addition, each chapter opens with an introductory vignette that sets the stage for some application of linear algebra and provides a motivation for developing the mathematics that follows. A Strong Geometric Emphasis Every major concept in the course is given a geometric interpretation, because many stu- dents learn better when they can visualize an idea. There are substantially more drawings here than usual, and some of the figures have never before appeared in a linear algebra text. Interactive versions of many of these figures appear in MyLab Math. Examples This text devotes a larger proportion of its expository material to examples than do most linear algebra texts. There are more examples than an instructor would ordinarily present in class. But because the examples are written carefully, with lots of detail, students can read them on their own.  Preface 15 Theorems and Proofs Important results are stated as theorems. Other useful facts are displayed in tinted boxes, for easy reference. Most of the theorems have formal proofs, written with the beginner student in mind. In a few cases, the essential calculations of a proof are exhibited in a carefully chosen example. Some routine verifications are saved for exercises, when they will benefit students. Practice Problems A few carefully selected Practice Problems appear just before each exercise set. Com- plete solutions follow the exercise set. These problems either focus on potential trouble spots in the exercise set or provide a “warm-up” for the exercises, and the solutions often contain helpful hints or warnings about the homework. Exercises The abundant supply of exercises ranges from routine computations to conceptual ques- tions that require more thought. A good number of innovative questions pinpoint con- ceptual difficulties that we have found on student papers over the years. Each exercise set is carefully arranged in the same general order as the text; homework assignments are readily available when only part of a section is discussed. A notable feature of the exercises is their numerical simplicity. Problems “unfold” quickly, so students spend little time on numerical calculations. The exercises concentrate on teaching understand- ing rather than mechanical calculations. The exercises in the Sixth Edition maintain the integrity of the exercises from previous editions, while providing fresh problems for students and instructors. Exercises marked with the symbol T are designed to be worked with the aid of a “matrix program” (a computer program, such as MATLAB, Maple, Mathematica, MathCad, or Derive, or a programmable calculator with matrix capabilities, such as those manufactured by Texas Instruments). True/False Questions To encourage students to read all of the text and to think critically, we have developed over 300 simple true/false questions that appear throughout the text, just after the com- putational problems. They can be answered directly from the text, and they prepare students for the conceptual problems that follow. Students appreciate these questions- after they get used to the importance of reading the text carefully. Based on class testing and discussions with students, we decided not to put the answers in the text. (The Study Guide, in MyLab Math, tells the students where to find the answers to the odd-numbered questions.) An additional 150 true/false questions (mostly at the ends of chapters) test understanding of the material. The text does provide simple T/F answers to most of these supplementary exercises, but it omits the justifications for the answers (which usually require some thought). Writing Exercises An ability to write coherent mathematical statements in English is essential for all stu- dents of linear algebra, not just those who may go to graduate school in mathematics. 16 Preface The text includes many exercises for which a written justification is part of the answer. Conceptual exercises that require a short proof usually contain hints that help a student get started. For all odd-numbered writing exercises, either a solution is included at the back of the text or a hint is provided and the solution is given in the Study Guide. Projects A list of projects (available online) have been identified at the end of each chapter. They can be used by individual students or in groups. These projects provide the opportunity for students to explore fundamental concepts and applications in more detail. Two of the projects even encourage students to engage their creative side and use linear transforma- tions to build artwork. Reasonable Answers Many of our students will enter a workforce where important decisions are being made based on answers provided by computers and other machines. The Reasonable Answers boxes and exercises help students develop an awareness of the need to analyze their answers for correctness and accuracy. Computational Topics The text stresses the impact of the computer on both the development and practice of linear algebra in science and engineering. Frequent Numerical Notes draw attention to issues in computing and distinguish between theoretical concepts, such as matrix inversion, and computer implementations, such as LU factorizations. Acknowledgments David Lay was grateful to many people who helped him over the years with various aspects of this book. He was particularly grateful to Israel Gohberg and Robert Ellis for more than fifteen years of research collaboration, which greatly shaped his view of linear algebra. And he was privileged to be a member of the Linear Algebra Curriculum Study Group along with David Carlson, Charles Johnson, and Duane Porter. Their creative ideas about teaching linear algebra have influenced this text in significant ways. He often spoke fondly of three good friends who guided the development of the book nearly from the beginning—giving wise counsel and encouragement—Greg Tobin, publisher; Laurie Rosatone, former editor; and William Hoffman, former editor. Judi and Steven have been privileged to work on recent editions of Professor David Lay’s linear algebra book. In making this revision, we have attempted to maintain the basic approach and the clarity of style that has made earlier editions popular with students and faculty. We thank Eric Schulz for sharing his considerable technological and peda- gogical expertise in the creation of the electronic textbook. His help and encouragement were essential in bringing the figures and examples to life in the Wolfram Cloud version of this textbook. Mathew Hudelson has been a valuable colleague in preparing the Sixth Edition; he is always willing to brainstorm about concepts or ideas and test out new writing and exercises. He contributed the idea for new vignette for Chapter 3 and the accompanying  Preface 17 project. He has helped with new exercises throughout the text. Harley Weston has pro- vided Judi with many years of good conversations about how, why, and who we appeal to when we present mathematical material in different ways. Katerina Tsatsomeros’ artistic side has been a definite asset when we needed artwork to transform (the fish and the sheep), improved writing in the new introductory vignettes, or information from the perspective of college-age students. We appreciate the encouragement and shared expertise from Nella Ludlow, Thomas Fischer, Amy Johnston, Cassandra Seubert, and Mike Manzano. They provided infor- mation about important applications of linear algebra and ideas for new examples and exercises. In particular, the new vignettes and material in Chapters 4 and 6 were inspired by conversations with these individuals. We are energized by Sepideh Stewart and the other new Linear Algebra Curricu- lum Study Group (LACSG 2.0) members: Sheldon Axler, Rob Beezer, Eugene Boman, Minerva Catral, Guershon Harel, David Strong, and Megan Wawro. Initial meetings of this group have provided valuable guidance in revising the Sixth Edition. We sincerely thank the following reviewers for their careful analyses and construc- tive suggestions: Maila C. Brucal-Hallare, Norfolk State University Steven Burrow, Central Texas College Kristen Campbell, Elgin Community College J. S. Chahal, Brigham Young University Charles Conrad, Volunteer State Community College Kevin Farrell, Lyndon State College R. Darrell Finney, Wilkes Community College Chris Fuller, Cumberland University Xiaofeng Gu, University of West Georgia Jeffrey Jauregui, Union College Jeong Mi-Yoon, University of Houston–Downtown Christopher Murphy, Guilford Tech. C.C. Michael T. Muzheve, Texas A&M U.–Kingsville Charles I. Odion, Houston Community College Iason Rusodimos, Perimeter C. at Georgia State U. Desmond Stephens, Florida Ag. and Mech. U. Rebecca Swanson, Colorado School of Mines Jiyuan Tao, Loyola University–Maryland Casey Wynn, Kenyon College Amy Yielding, Eastern Oregon University Taoye Zhang, Penn State U.–Worthington Scranton Houlong Zhuang, Arizona State University We appreciate the proofreading and suggestions provided by John Samons and Jennifer Blue. Their careful eye has helped to minimize errors in this edition. We thank Kristina Evans, Phil Oslin, and Jean Choe for their work in setting up and maintaining the online homework to accompany the text in MyLab Math, and for continuing to work with us to improve it. The reviews of the online homework done by Joan Saniuk, Robert Pierce, Doron Lubinsky and Adriana Corinaldesi were greatly appreciated. We also thank the faculty at University of California Santa Barbara, Uni- versity of Alberta, Washington State University and the Georgia Institute of Technology for their feedback on the MyLab Math course. Joe Vetere has provided much appreciated technical help with the Study Guide and Instructor’s Solutions Manual. We thank Jeff Weidenaar, our content manager, for his continued careful, well- thought-out advice. Project Manager Ron Hampton has been a tremendous help guiding us through the production process. We are also grateful to Stacey Sveum and Rosemary Morton, our marketers, and Jon Krebs, our editorial associate, who have also contributed to the success of this edition. Steven R. Lay and Judi J. McDonald 18 Preface Acknowledgments for the Global Edition Pearson would like to acknowledge and thank the following for their work on the Global Edition. Contributors José Luis Zuleta Estrugo, École Polytechnique Fédérale de Lausanne Mohamad Rafi Segi Rahmat, University of Nottingham Malaysia Reviewers Sibel Doğru Akgöl, Atilim University Hossam M. Hassan, Cairo University Kwa Kiam Heong, University of Malaya Yanghong Huang, University of Manchester Natanael Karjanto, Sungkyunkwan University Somitra Sanadhya, Indraprastha Institute of Information Technology Veronique Van Lierde, Al Akhawayn University in Ifrane  Get the most out of MyLab Math MyLab Math for Linear Algebra and Its Applications Lay, Lay, McDonald MyLab Math features hundreds of assignable algorithmic exercises that mirror range of author-created resources, so your students have a consistent experience. eText with Interactive Figures The eText includes Interactive Figures that bring the geometry of linear algebra to life. Students can manipulate with matrices to provide a deeper geometric understanding of key concepts and examples. Teaching with Interactive Figures as a teaching tool for classroom demonstrations. Instructors can illustrate concep for students to visualize, leading to greater conceptual understanding. pearson.com/mylab/math Supporting Instruction MyLab Math provides resources to help you assess and improve student results and unparalleled flexibility to create a course tailored to you and your students. PowerPoint® Lecture Slides Fully editable PowerPoint slides are available for all sections of the text. The slides include definitions, theorems, examples and solutions. When used in the classroom, these slides allow the instructor to focus on teaching, rather than writing on the board. PowerPoint slides are available to students (within the Video and Resource Library in MyLab Math) so that they can follow along. Sample Assignments Sample Assignments are crafted to maximize student performance in the course. They make course set-up easier by giving instructors a starting point for each section. Comprehensive Gradebook The gradebook includes enhanced reporting functionality, such as item analysis and a reporting dashboard to course. Student performance data are presented at the class, section, and program levels in an accessible, visual manner so you’ll have the information you need to keep your students on track. pearson.com/mylab/math  Resources for Success Instructor Resources Student Resources Online resources can be downloaded Additional resources to enhance from MyLab Math or from student success. All resources can be www.pearsonglobaleditions.com. downloaded from MyLab Math. Instructor’s Solution Manual Study Guide Includes fully worked solutions to all exercises in Provides detailed worked-out solutions to the text and teaching notes for many sections. every third odd-numbered exercise. Also, a complete explanation is provided whenever PowerPoint® Lecture Slides an odd-numbered writing exercise has a These fully editable lecture slides are available Hint in the answers. Special subsections of for all sections of the text. the Study Guide suggest how to master key concepts of the course. Frequent “Warnings” identify common errors and show how to Instructor’s Technology Manuals prevent them. MATLAB boxes introduce Each manual provides detailed guidance for commands as they are needed. Appendixes integrating technology throughout the course, in the Study Guide provide comparable infor- written by faculty who teach with the software mation about Maple, Mathematica, and TI and this text. Available For MATLAB, Maple, graphing calculators. Available within MyLab Mathematica, and Texas Instruments graphing math. calculators. TestGen® Getting Started with Technology TestGen (www.pearsoned.com/testgen) enables A quick-start guide for students to the tech- instructors to build, edit, print, and administer nology they may use in this course. Available tests using a computerized bank of questions for MATLAB, Maple, Mathematica, or Texas developed to cover all the objectives of the text. Instrument graphing calculators. Downloadable from MyLab Math. Projects Exploratory projects, written by experienced faculty members, invite students to discover applications of linear algebra. pearson.com/mylab/math A Note to Students This course is potentially the most interesting and worthwhile undergraduate mathe- matics course you will complete. In fact, some students have written or spoken to us after graduation and said that they still use this text occasionally as a reference in their careers at major corporations and engineering graduate schools. The following remarks offer some practical advice and information to help you master the material and enjoy the course. In linear algebra, the concepts are as important as the computations. The simple numerical exercises that begin each exercise set only help you check your understanding of basic procedures. Later in your career, computers will do the calculations, but you will have to choose the calculations, know how to interpret the results, analyze whether the results are reasonable, then explain the results to other people. For this reason, many exercises in the text ask you to explain or justify your calculations. A written explanation is often required as part of the answer. If you are working on questions in MyLab Math, keep a notebook for calculations and notes on what you are learning. For odd-numbered exercises in the textbook, you will find either the desired explanation or at least a good hint. You must avoid the temptation to look at such answers before you have tried to write out the solution yourself. Otherwise, you are likely to think you understand something when in fact you do not. To master the concepts of linear algebra, you will have to read and reread the text carefully. New terms are in boldface type, sometimes enclosed in a definition box. A glossary of terms is included at the end of the text. Important facts are stated as theorems or are enclosed in tinted boxes, for easy reference. We encourage you to read the Preface to learn more about the structure of this text. This will give you a framework for understanding how the course may proceed. In a practical sense, linear algebra is a language. You must learn this language the same way you would a foreign language—with daily work. Material presented in one section is not easily understood unless you have thoroughly studied the text and worked the exercises for the preceding sections. Keeping up with the course will save you lots of time and distress! Numerical Notes We hope you read the Numerical Notes in the text, even if you are not using a computer or graphing calculator with the text. In real life, most applications of linear algebra involve numerical computations that are subject to some numerical error, even though that error may be extremely small. The Numerical Notes will warn you of potential difficulties in 22  A Note to Students 23 using linear algebra later in your career, and if you study the notes now, you are more likely to remember them later. If you enjoy reading the Numerical Notes, you may want to take a course later in numerical linear algebra. Because of the high demand for increased computing power, computer scientists and mathematicians work in numerical linear algebra to develop faster and more reliable algorithms for computations, and electrical engineers design faster and smaller computers to run the algorithms. This is an exciting field, and your first course in linear algebra will help you prepare for it. Study Guide To help you succeed in this course, we suggest that you use the Study Guide available in MyLab Math. Not only will it help you learn linear algebra, it also will show you how to study mathematics. At strategic points in your textbook, marginal notes will remind you to check that section of the Study Guide for special subsections entitled “Mastering Linear Algebra Concepts.” There you will find suggestions for constructing effective review sheets of key concepts. The act of preparing the sheets is one of the secrets to success in the course, because you will construct links between ideas. These links are the “glue” that enables you to build a solid foundation for learning and remembering the main concepts in the course. The Study Guide contains a detailed solution to more than a third of the odd- numbered exercises, plus solutions to all odd-numbered writing exercises for which only a hint is given in the Answers section of this book. The Guide is separate from the text because you must learn to write solutions by yourself, without much help. (We know from years of experience that easy access to solutions in the back of the text slows the mathematical development of most students.) The Guide also provides warnings of common errors and helpful hints that call attention to key exercises and potential exam questions. If you have access to technology—MATLAB, Octave, Maple, Mathematica, or a TI graphing calculator—you can save many hours of homework time. The Study Guide is your “lab manual” that explains how to use each of these matrix utilities. It introduces new commands when they are needed. You will also find that most software commands you might use are easily found using an online search engine. Special matrix commands will perform the computations for you! What you do in your first few weeks of studying this course will set your pattern for the term and determine how well you finish the course. Please read “How to Study Linear Algebra” in the Study Guide as soon as possible. Many students have found the strategies there very helpful, and we hope you will, too. This page is intentionally left blank  1 Linear Equations in Linear Algebra Introductory Example LINEAR MODELS IN ECONOMICS AND ENGINEERING It was late summer in 1949. Harvard Professor Wassily mathematical model. Since that time, researchers in Leontief was carefully feeding the last of his punched cards many other fields have employed computers to analyze into the university’s Mark II computer. The cards contained mathematical models. Because of the massive amounts of information about the U.S. economy and represented a data involved, the models are usually linear; that is, they summary of more than 250,000 pieces of information are described by systems of linear equations. produced by the U.S. Bureau of Labor Statistics after two The importance of linear algebra for applications has years of intensive work. Leontief had divided the U.S. risen in direct proportion to the increase in computing economy into 500 “sectors,” such as the coal industry, power, with each new generation of hardware and software the automotive industry, communications, and so on. triggering a demand for even greater capabilities. Computer For each sector, he had written a linear equation that science is thus intricately linked with linear algebra through described how the sector distributed its output to the other the explosive growth of parallel processing and large-scale sectors of the economy. Because the Mark II, one of the computations. largest computers of its day, could not handle the resulting system of 500 equations in 500 unknowns, Leontief had Scientists and engineers now work on problems far distilled the problem into a system of 42 equations in more complex than even dreamed possible a few decades 42 unknowns. ago. Today, linear algebra has more potential value for students in many scientific and business fields than any Programming the Mark II computer for Leontief’s other undergraduate mathematics subject! The material in 42 equations had required several months of effort, and he this text provides the foundation for further work in many was anxious to see how long the computer would take to interesting areas. Here are a few possibilities; others will solve the problem. The Mark II hummed and blinked for be described later. 56 hours before finally producing a solution. We will discuss the nature of this solution in Sections 1.6 and 2.6.  Oil exploration. When a ship searches for offshore Leontief, who was awarded the 1973 Nobel Prize oil deposits, its computers solve thousands of in Economic Science, opened the door to a new era separate systems of linear equations every day. in mathematical modeling in economics. His efforts at The seismic data for the equations are obtained Harvard in 1949 marked one of the first significant uses from underwater shock waves created by explosions of computers to analyze what was then a large-scale from air guns. The waves bounce off subsurface 25 26 CHAPTER 1 Linear Equations in Linear Algebra rocks and are measured by geophones attached to relies on linear algebra techniques and systems of mile-long cables behind the ship. linear equations.  Linear programming. Many important management  Artificial intelligence. Linear algebra plays a key decisions today are made on the basis of linear role in everything from scrubbing data to facial programming models that use hundreds of recognition. variables. The airline industry, for instance, employs  Signals and signal processing. From a digital linear programs that schedule flight crews, monitor photograph to the daily price of a stock, important the locations of aircraft, or plan the varied schedules information is recorded as a signal and processed of support services such as maintenance and using linear transformations. terminal operations.  Machine learning. Machines (specifically comput-  Electrical networks. Engineers use simulation ers) use linear algebra to learn about anything from software to design electrical circuits and microchips online shopping preferences to speech recognition. involving millions of transistors. Such software Systems of linear equations lie at the heart of linear algebra, and this chapter uses them to introduce some of the central concepts of linear algebra in a simple and concrete setting. Sections 1.1 and 1.2 present a systematic method for solving systems of linear equations. This algorithm will be used for computations throughout the text. Sections 1.3 and 1.4 show how a system of linear equations is equivalent to a vector equation and to a matrix equation. This equivalence will reduce problems involving linear combinations of vectors to questions about systems of linear equations. The fundamental concepts of spanning, linear independence, and linear transformations, studied in the second half of the chapter, will play an essential role throughout the text as we explore the beauty and power of linear algebra. 1.1 Systems of Linear Equations A linear equation in the variables x1 ; : : : ; xn is an equation that can be written in the form a1 x1 C a2 x2 C


C an xn D b (1) where b and the coefficients a1 ; : : : ; an are real or complex numbers, usually known in advance. The subscript n may be any positive integer. In textbook examples and exercises, n is normally between 2 and 5. In real-life problems, n might be 50 or 5000, or even larger. The equations p
4x1 5x2 C 2 D x1 and x2 D 2 6 x1 C x3 are both linear because they can be rearranged algebraically as in equation (1): p 3x1 5x2 D 2 and 2x1 C x2 x3 D 2 6 The equations p 4x1 5x2 D x1 x2 and x2 D 2 x1 6 p are not linear because of the presence of x1 x2 in the first equation and x1 in the second.  1.1 Systems of Linear Equations 27 A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables—say, x1 ; : : : ; xn. An example is 2x1 x2 C 1:5x3 D 8 (2) x1 4x3 D 7 A solution of the system is a list.s1 ; s2 ; : : : ; sn / of numbers that makes each equation a true statement when the values s1 ; : : : ; sn are substituted for x1 ; : : : ; xn , respectively. For instance,.5; 6:5; 3/ is a solution of system (2) because, when these values are substituted in (2) for x1 ; x2 ; x3 , respectively, the equations simplify to 8 D 8 and 7 D 7. The set of all possible solutions is called the solution set of the linear system. Two linear systems are called equivalent if they have the same solution set. That is, each solution of the first system is a solution of the second system, and each solution of the second system is a solution of the first. Finding the solution set of a system of two linear equations in two variables is easy because it amounts to finding the intersection of two lines. A typical problem is x1 2x2 D 1 x1 C 3x2 D 3 The graphs of these equations are lines, which we denote by `1 and `2. A pair of numbers.x1 ; x2 / satisfies both equations in the system if and only if the point.x1 ; x2 / lies on both `1 and `2. In the system above, the solution is the single point.3; 2/, as you can easily verify. See Figure 1. x2 2 x1 3 /2 /1 FIGURE 1 Exactly one solution. Of course, two lines need not intersect in a single point—they could be parallel, or they could coincide and hence “intersect” at every point on the line. Figure 2 shows the graphs that correspond to the following systems: (a) x1 2x2 D 1 (b) x1 2x2 D 1 x1 C 2x2 D 3 x1 C 2x2 D 1 Figures 1 and 2 illustrate the following general fact about linear systems, to be verified in Section 1.2. x2 x2 2 2 x1 x1 3 3 /2 /1 /1 (a) (b) FIGURE 2 (a) No solution. (b) Infinitely many solutions. 28 CHAPTER 1 Linear Equations in Linear Algebra A system of linear equations has 1. no solution, or 2. exactly one solution, or 3. infinitely many solutions. A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution. Matrix Notation The essential information of a linear system can be recorded compactly in a rectangular array called a matrix. Given the system x1 2x2 C x3 D 0 2x2 8x3 D 8 (3) 5x1 5x3 D 10 with the coefficients of each variable aligned in columns, the matrix 2 3 1 2 1 40 2 85 5 0 5 is called the coefficient matrix (or matrix of coefficients) of the system (3), and the matrix 2 3 1 2 1 0 40 2 8 85 (4) 5 0 5 10 is called the augmented matrix of the system. (The second row here contains a zero because the second equation could be written as 0
x1 C 2x2 8x3 D 8.) An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the respective right sides of the equations. The size of a matrix tells how many rows and columns it has. The augmented matrix (4) above has 3 rows and 4 columns and is called a 3  4 (read “3 by 4”) matrix. If m and n are positive integers, an m  n matrix is a rectangular array of numbers with m rows and n columns. (The number of rows always comes first.) Matrix notation will simplify the calculations in the examples that follow. Solving a Linear System This section and the next describe an algorithm, or a systematic procedure, for solving linear systems. The basic strategy is to replace one system with an equivalent system (that is one with the same solution set) that is easier to solve. Roughly speaking, use the x1 term in the first equation of a system to eliminate the x1 terms in the other equations. Then use the x2 term in the second equation to eliminate the x2 terms in the other equations, and so on, until you finally obtain a very simple equivalent system of equations.  1.1 Systems of Linear Equations 29 Three basic operations are used to simplify a linear system: Replace one equation by the sum of itself and a multiple of another equation, interchange two equations, and multiply all the terms in an equation by a nonzero constant. After the first example, you will see why these three operations do not change the solution set of the system. EXAMPLE 1 Solve system (3). SOLUTION The elimination procedure is shown here with and without matrix nota- tion, and the results are placed side by side for comparison: 2 3 x1 2x 2 C x3 D 0 1 2 1 0 2x2 8x3 D 8 40 2 8 85 5x 5x D 10 5 0 5 10 1 3 Keep x1 in the first equation and eliminate it from the other equations. To do so, add 5 times equation 1 to equation 3. After some practice, this type of calculation is usually performed mentally: 5
Œequation 1 5x1 C 10x 2 5x3 D 0 C Œequation 3 5x1 5x3 D 10 Œnew equation 3 10x 2 10x3 D 10 The result of this calculation is written in place of the original third equation: 2 3 x1 2x 2 C x3 D 0 1 2 1 0 2x2 8x3 D 8 40 2 8 85 10x 10x D 10 0 10 10 10 2 3 Now, multiply equation 2 by 12 in order to obtain 1 as the coefficient for x2. (This calculation will simplify the arithmetic in the next step.) 2 3 x1 2x 2 C x3 D 0 1 2 1 0 x2 4x3 D 4 40 1 4 45 10x 10x D 10 0 10 10 10 2 3 Use the x2 in equation 2 to eliminate the 10x2 in equation 3. The “mental’’ computation is 10
Œequation 2 10x 2 C 40x3 D 40 C Œequation 3 10x 2 10x3 D 10 Œnew equation 3 30x3 D 30 The result of this calculation is written in place of the previous third equation (row): 2 3 x1 2x 2 C x3 D 0 1 2 1 0 x2 4x3 D 4 40 1 4 45 30x D 30 0 0 30 30 3 1 Now, multiply equation 3 by 30 in order to obtain 1 as the coefficient for x3. (This calculation will simplify the arithmetic in the next step.) 2 3 x1 2x 2 C x3 D 0 1 2 1 0 x2 4x3 D 4 4 0 1 4 45 x3 D 1 0 0 1 1 30 CHAPTER 1 Linear Equations in Linear Algebra The new system has a triangular form (the intuitive term triangular will be replaced by a precise term in the next section): 2 3 x1 2x 2 C x3 D 0 1 2 1 0 x2 4x3 D 4 40 1 4 45 x D 1 0 0 1 1 3 Eventually, you want to eliminate the 2x2 term from equation 1, but it is more efficient to use the x3 in equation 3 first, to eliminate the 4x3 and Cx3 terms in equations 2 and 1. The two “mental” calculations are 4
Œequation 3 4x3 D 4 1
Œequation 3 x3 D 1 C Œequation 2 x2 4x3 D 4 C Œequation 1 x1 2x2 C x3 D 0 Œnew equation 2 x2 D 0 Œnew equation 1 x1 2x2 D1 x3 It is convenient to combine the results of these two operations: 2 3 x1 2x 2 D 1 1 2 0 1 (1, 0, 21) x2 D 0 40 1 0 05 x D 1 0 0 1 1 3 Now, having cleaned out the column above the x3 in equation 3, move back to the x2 in equation 2 and use it to eliminate the 2x2 above it. Because of the previous work with x2 x3 , there is now no arithmetic involving x3 terms. Add 2 times equation 2 to equation 1 x1 and obtain the system: 2 3 x1 D 1 1 0 0 1 Each of the original equations x2 D 0 40 1 0 05 determines a plane in 0 0 1 1 three-dimensional space. The point x D 1 3.1; 0; 1/ lies in all three planes. The work is essentially done. It shows that the only solution of the original system is.1; 0; 1/. However, since there are so many calculations involved, it is a good practice to check the work. To verify that.1; 0; 1/ is a solution, substitute these values into the left side of the original system, and compute: 1.1/ 2.0/ C 1. 1/ D 1 0 1D 0 2.0/ 8. 1/ D 0C8D 8 5.1/ 5. 1/ D 5 C 5 D 10 The results agree with the right side of the original system, so.1; 0; 1/ is a solution of the system. Example 1 illustrates how operations on equations in a linear system correspond to operations on the appropriate rows of the augmented matrix. The three basic operations listed earlier correspond to the following operations on the augmented matrix. ELEMENTARY ROW OPERATIONS 1. (Replacement) Replace one row by the sum of itself and a multiple of another row.1 2. (Interchange) Interchange two rows. 3. (Scaling) Multiply all entries in a row by a nonzero constant. 1 A common paraphrase of row replacement is “Add to one row a multiple of another row.”  1.1 Systems of Linear Equations 31 Row operations can be applied to any matrix, not merely to one that arises as the augmented matrix of a linear system. Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. It is important to note that row operations are reversible. If two rows are inter- changed, they can be returned to their original positions by another interchange. If a row is scaled by a nonzero constant c , then multiplying the new row by 1=c produces the original row. Finally, consider a replacement operation involving two rows—say, rows 1 and 2—and suppose that c times row 1 is added to row 2 to produce a new row 2. To “reverse” this operation, add c times row 1 to (new) row 2 and obtain the original row 2. See Exercises 39–42 at the end of this section. At the moment, we are interested in row operations on the augmented matrix of a system of linear equations. Suppose a system is changed to a new one via row operations. By considering each type of row operation, you can see that any solution of the original system remains a solution of the new system. Conversely, since the original system can be produced via row operations on the new system, each solution of the new system is also a solution of the original system. This discussion justifies the following statement. If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. Though Example 1 is lengthy, you will find that after some practice, the calculations go quickly. Row operations in the text and exercises will usually be extremely easy to perform, allowing you to focus on the underlying concepts. Still, you must learn to perform row operations accurately because they will be used throughout the text. The rest of this section shows how to use row operations to determine the size of a solution set, without completely solving the linear system. Existence and Uniqueness Questions Section 1.2 will show why a solution set for a linear system contains either no solutions, one solution, or infinitely many solutions. Answers to the following two questions will determine the nature of the solution set for a linear system. To determine which possibility is true for a particular system, we ask two questions. TWO FUNDAMENTAL QUESTIONS ABOUT A LINEAR SYSTEM 1. Is the system consistent; that is, does at least one solution exist? 2. If a solution exists, is it the only one; that is, is the solution unique? These two questions will appear throughout the text, in many different guises. This section and the next will show how to answer these questions via row operations on the augmented matrix. EXAMPLE 2 Determine if the following system is consistent: x1 2x2 C x3 D 0 2x2 8x3 D 8 5x1 5x3 D 10 32 CHAPTER 1 Linear Equations in Linear Algebra SOLUTION This is the system from Example 1. Suppose that we have performed the row operations necessary to obtain the triangular form 2 3 x1 2x2 C x3 D 0 1 2 1 0 x2 4x3 D 4 40 1 4 45 x D 1 0 0 1 1 3 At this point, we know x3. Were we to substitute the value of x3 into equation 2, we could compute x2 and hence could determine x1 from equation 1. So a solution exists; the system is consistent. (In fact, x2 is uniquely determined by equation 2 since x3 has only one possible value, and x1 is therefore uniquely determined by equation 1. So the solution is unique.) EXAMPLE 3 Determine if the following system is consistent: x2 4x3 D 8 2x1 3x2 C 2x3 D 1 (5) 4x1 8x2 C 12x3 D 1 SOLUTION The augmented matrix is 2 3 0 1 4 8 42 3 2 15 4 8 12 1 To obtain an x1 in the first equation, interchange rows 1 and 2: 2 3 2 3 2 1 40 1 4 85 4 8 12 1 To eliminate the 4x1 term in the third equation, add 2 times row 1 to row 3: 2 3 2 3 2 1 40 1 4 85 (6) 0 2 8 1 Next, use the x2 term in the second equation to eliminate the 2x2 term from the third equation. Add 2 times row 2 to row 3: 2 3 x3 2 3 2 1 40 1 4 85 (7) 0 0 0 15 The augmented matrix is now in triangular form. To interpret it correctly, go back to equation notation: x2 2x1 3x2 C 2x3 D 1 x1 (8) x2 4x3 D 8 0 D 15 The equation 0 D 15 is a short form of 0x1 C 0x2 C 0x3 D 15. This system in trian- The system is inconsistent because gular form obviously has a built-in contradiction. There are no values of x1 ; x2 ; x3 that there is no point that lies on all satisfy (8) because the equation 0 D 15 is never true. Since (8) and (5) have the same three planes. solution set, the original system is inconsistent (it has no solution). Pay close attention to the augmented matrix in (7). Its last row is typical of an inconsistent system in triangular form.  1.1 Systems of Linear Equations 33 Reasonable Answers Once you have one or more solutions to a system of equations, remember to check your answer by substituting the solution you found back into the original equation. For example, if you found.2; 1; 1/ was a solution to the system of equations x1 2x2 C x3 D 2 x1 2x3 D 2 x2 C x3 D 3 you could substitute your solution into the original equations to get 2 2.1/ C. 1/ D 1 ¤ 2 2 2. 1/ D 4 ¤ 2 1 C. 1/ D 0 ¤ 3 It is now clear that there must have been an error in your original calculations. If upon rechecking your arithmetic, you find the answer.2; 1; 2/, you can see that 2 2.1/ C.2/ D 2 D 2 2 2.2/ D 2 D 2 1 C 2 D 3 D 3 and you can now be confident you have a correct solution to the given system of equations. Numerical Note In real-world problems, systems of linear equations are solved by a computer. For a square coefficient matrix, computer programs nearly always use the elim- ination algorithm given here and in Section 1.2, modified slightly for improved accuracy. The vast majority of linear algebra problems in business and industry are solved with programs that use floating point arithmetic. Numbers are represented as decimals ˙:d1


dp  10r , where r is an integer and the number p of digits to the right of the decimal point is usually between 8 and 16. Arithmetic with such numbers typically is inexact, because the result must be rounded (or truncated) to the number of digits stored. “Roundoff error” is also introduced when a number such as 1=3 is entered into the computer, since its decimal representation must be approximated by a finite number of digits. Fortunately, inaccuracies in floating point arithmetic seldom cause problems. The numerical notes in this book will occasionally warn of issues that you may need to consider later in your career. Practice Problems Throughout the text, practice problems should be attempted before working the exer- cises. Solutions appear after each exercise set. 1. State in words the next elementary row operation that should be performed on the system in order to solve it. [More than one answer is possible in (a).] 34 CHAPTER 1 Linear Equations in Linear Algebra Practice Problems (Continued) a. x1 C 4x2 2x3 C 8x4 D 12 b. x1 3x2 C 5x3 2x4 D 0 x2 7x3 C 2x4 D 4 x2 C 8x3 D 4 5x3 x4 D 7 2x3 D 3 x3 C 3x4 D 5 x4 D 1 2. The augmented matrix of a linear system has been transformed by row operations into the form below. Determine if the system is consistent. 2 3 1 5 2 6 40 4 7 25 0 0 5 0 3. Is.3; 4; 2/ a solution of the following system? 5x1 x2 C 2x3 D 7 2x1 C 6x2 C 9x3 D 0 7x1 C 5x2 3x3 D 7 4. For what values of h and k is the following system consistent? 2x1 x