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1 Linear Algebra with Applications This page intentionally left blank Linear Algebra with Applications Fifth Edition Otto Bretscher Colby College Editor in Chief: Christine Hoag Cover Designer: Suzanne Duda Senior Acquisitions Editor: William Hoffman Cover Art: Colorization by Jorgensen Executive Marketing Manager: Jeff Weidenaar Fernandez/NASA. Marketing Assistant: Caitlin Crain Full-Service Project Management: Senior Production Project Manager: Integra Software Services, Ltd. Beth Houston Composition: Integra Software Services, Ltd. Manager, Cover Visual Printer/Binder: Edwards Brothers Malloy Research and Permissions: Jayne Conte Cover Printer: Lehigh/Phoenix The cover shows the Mars rover Curiosity, casting a long shadow onto Gale crater, facing Aeolis Mons. Linear Algebra plays a central role in many aspects of the planning, design, and control of a space mission. For example, data compression is used for interplanetary communication (see Page 411), and error-correction codes increase the reliability of data transmission (see Page 121).

2 Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on the appropriate page within text. Copyright 2013, 2009, 2005 by Pearson Education, Inc. All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458, or you may fax your request to 201-236-3290. Many of the designations by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps.

3 Library of Congress Cataloging-in-Publication Data Bretscher, Otto. Linear Algebra with Applications / Otto Bretscher. 5th ed. p. cm. Includes index. ISBN-13: 978-0-321-79697-4. ISBN-10: 0-321-79697-7. 1. Algebras, Linear Textbooks. I. Title. 2013. 512'.5 dc23. 2012017551. 10 9 8 7 6 5 4 3 2 1 EBM 16 15 14 13 12. ISBN-10: 0-321-79697-7. ISBN-13: 978-0-321-79697-4. To my parents Otto and Margrit Bretscher-Zwicky with love and gratitude This page intentionally left blank Contents Preface ix 1 Linear Equations 1. Introduction to Linear Systems 1. Matrices, Vectors, and Gauss Jordan Elimination 8. On the Solutions of Linear Systems; Matrix Algebra 25. 2 Linear Transformations 41. Introduction to Linear Transformations and Their Inverses 41. Linear Transformations in Geometry 58. Matrix Products 75. The Inverse of a Linear Transformation 88. 3 Subspaces of Rn and Their Dimensions 110. Image and Kernel of a Linear Transformation 110. Subspaces of Rn ; Bases and Linear Independence 122.

4 The Dimension of a Subspace of Rn 133. Coordinates 147. 4 Linear Spaces 166. Introduction to Linear Spaces 166. Linear Transformations and Isomorphisms 178. The Matrix of a Linear Transformation 186. 5 Orthogonality and Least Squares 202. Orthogonal Projections and Orthonormal Bases 202. Gram Schmidt Process and QR Factorization 218. Orthogonal Transformations and Orthogonal Matrices 225. Least Squares and Data Fitting 236. Inner Product Spaces 249. vii viii Contents 6 Determinants 265. Introduction to Determinants 265. Properties of the Determinant 277. Geometrical Interpretations of the Determinant;. Cramer's Rule 294. 7 eigenvalues and eigenvectors 310. Diagonalization 310. Finding the Eigenvalues of a Matrix 327. Finding the eigenvectors of a Matrix 339. More on Dynamical Systems 347. Complex Eigenvalues 360. Stability 375. 8 Symmetric Matrices and Quadratic Forms 385. Symmetric Matrices 385. Quadratic Forms 394. Singular Values 403. 9 Linear Differential Equations 415.

5 An Introduction to Continuous Dynamical Systems 415. The Complex Case: Euler's Formula 429. Linear Differential Operators and Linear Differential Equations 442. Appendix A Vectors 457. Appendix B Techniques of Proof 467. Answers to Odd-Numbered Exercises 471. Subject Index 499. Name Index 507. Preface (with David Steinsaltz). A. police of cer on patrol at midnight, so runs an old joke, notices a man crawling about on his hands and knees under a streetlamp. He walks over to investigate, whereupon the man explains in a tired and somewhat slurred voice that he has lost his housekeys. The policeman offers to help, and for the next ve minutes he too is searching on his hands and knees. At last he exclaims, Are you absolutely certain that this is where you dropped the keys? . Here? Absolutely not. I dropped them a block down, in the middle of the street.. Then why the devil have you got me hunting around this lamppost? . Because this is where the light is.

6 It is mathematics, and not just (as Bismarck claimed) politics, that consists in the art of the possible. Rather than search in the darkness for solutions to problems of pressing interest, we contrive a realm of problems whose interest lies above all in the fact that solutions can conceivably be found. Perhaps the largest patch of light surrounds the techniques of matrix arithmetic and Algebra , and in particular matrix multiplication and row reduction. Here we might begin with Descartes, since it was he who discovered the conceptual meeting- point of geometry and Algebra in the identi cation of Euclidean space with R3 ; the techniques and Applications proliferated since his day. To organize and clarify those is the role of a modern Linear Algebra course. Computers and Computation An essential issue that needs to be addressed in establishing a mathematical method- ology is the role of computation and of computing technology. Are the proper sub- jects of mathematics algorithms and calculations, or are they grand theories and abstractions that evade the need for computation?

7 If the former, is it important that the students learn to carry out the computations with pencil and paper, or should the algorithm press the calculator's x 1 button be allowed to substitute for the tradi- tional method of nding an inverse? If the latter, should the abstractions be taught through elaborate notational mechanisms or through computational examples and graphs? We seek to take a consistent approach to these questions: Algorithms and com- putations are primary, and precisely for this reason computers are not. Again and again we examine the nitty-gritty of row reduction or matrix multiplication in or- der to derive new insights. Most of the proofs, whether of rank-nullity theorem, the volume-change formula for determinants, or the spectral theorem for symmetric matrices, are in this way tied to hands-on procedures. The aim is not just to know how to compute the solution to a problem, but to imagine the computations. The student needs to perform enough row reductions by hand to be equipped to follow a line of argument of the form: If we calculate the reduced row-echelon form of such a matrix.

8 , and to appreciate in advance the possible outcomes of a particular computation. ix x Preface In Applications , the solution to a problem is hardly more important than recog- nizing its range of validity and appreciating how sensitive it is to perturbations of the input. We emphasize the geometric and qualitative nature of the solutions, notions of approximation, stability, and typical matrices. The discussion of Cramer's rule, for instance, underscores the value of closed-form solutions for visualizing a sys- tem's behavior and understanding its dependence on initial conditions. The availability of computers is, however, neither to be ignored nor regretted. Each student and instructor will have to decide how much practice is needed to be suf ciently familiar with the inner workings of the algorithm. As the explicit com- putations are being replaced gradually by a theoretical overview of how the algo- rithm works, the burden of calculation will be taken up by technology, particularly for those wishing to carry out the more numerical and applied exercises.

9 Examples, Exercises, Applications , and History The exercises and examples are the heart of this book. Our objective is not just to show our readers a patch of light where questions may be posed and solved, but to convince them that there is indeed a great deal of useful, interesting material to be found in this area if they take the time to look around. Consequently, we have included genuine Applications of the ideas and methods under discussion to a broad range of sciences: physics, computer science, chemistry, biology, economics, and, of course, mathematics itself. Often we have simpli ed them to sharpen the point, but they use the methods and models of contemporary scientists. With such a large and varied set of exercises in each section, instructors should have little dif culty in designing a course that is suited to their aims and to the needs of their students. Quite a few straightforward computation problems are offered, of course. Simple (and, in a few cases, not so simple) proofs and derivations are required in some exercises.

10 In many cases, theoretical principles that are discussed at length in more abstract Linear Algebra courses are here found broken up in bite- size exercises. The examples make up a signi cant portion of the text; we have kept abstract exposition to a minimum. It is a matter of taste whether general theories should give rise to speci c examples or be pasted together from them. In a text such as this one, attempting to keep an eye on Applications , the latter is clearly preferable: The examples always precede the theorems in this book. Scattered throughout the mathematical exposition are quite a few names and dates, some historical accounts, and anecdotes. Students of mathematics are too rarely shown that the seemingly strange and arbitrary concepts they study are the results of long and hard struggles. It will encourage the readers to know that a mere two centuries ago some of the most brilliant mathematicians were wrestling with problems such as the meaning of dimension or the interpretation of eit , and to realize that the advance of time and understanding actually enables them, with some effort of their own, to see farther than those great minds.