2012 • 490 Pages • 12.2 MB • English

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Contents Text Features ix Preface xi 1 Linear Equations 1 1.1 Introduction to Linear Systems 1 1.2 Matrices, Vectors, and Gauss-Jordan Elimination 8 1.3 On the Solutions of Linear Systems; Matrix Algebra 25 2 Linear Transformations 40 2.1 Introduction to Linear Transformations and Their Inverses 40 2.2 Linear Transformations in Geometry 54 2.3 Matrix Products 69 2.4 The Inverse of a Linear Transformation 79 3 Subspaces of Mn and Their Dimensions 101 3.1 Image and Kernel of a Linear Transformation 101 3.2 Subspaces of R"; Bases and Linear Independence 113 3.3 The Dimension of a Subspace of R" 123 3.4 Coordinates 137 4 Linear Spaces 153 4.1 Introduction to Linear Spaces 153 4.2 Linear Transformations and Isomorphisms 165 4.3 The Matrix of a Linear Transformation 112 5 Orthogonality and Least Squares 187 5.1 Orthogonal Projections and Orthonormal Bases 187 5.2 Gram-Schmidt Process and QR Factorization 203 5.3 Orthogonal Transformations and Orthogonal Matrices 210 5.4 Least Squares and Data Fitting 220 5.5 Inner Product Spaces 233 vii

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viii Contents 6 Determinants 249 6.1 Introduction to Determinants 249 6.2 Properties of the Determinant 261 6.3 Geometrical Interpretations of the Determinant; Cramer’s Rule 277 7 Eigenvalues and Eigenvectors 294 7.1 Dynamical Systems and Eigenvectors: An Introductory Example 294 7.2 Finding the Eigenvalues of a Matrix 308 7.3 Finding the Eigenvectors of a Matrix 319 7.4 Diagonalization 332 7.5 Complex Eigenvalues 343 7.6 Stability 357 8 Symmetric Matrices and Quadratic Forms 367 8.1 Symmetric Matrices 367 8.2 Quadratic Forms 376 8.3 Singular Values 385 9 Linear Differential Equations 397 9.1 An Introduction to Continuous Dynamical Systems 397 9.2 The Complex Case: Euler’s Formula 410 9.3 Linear Differential Operators and Linear Differential Equations 423 Appendix A Vectors 437 Answers to Odd-Numbered Exercises 446 Subject Index 471 Name Index 478

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Text Features Continuing Text Features • Linear transformations are introduced early on in the text to make the discus sion of matrix operations more meaningful and easier to visualize. • Visualization and geometrical interpretation are emphasized extensively throughout. • The reader will find an abundance of thought-provoking (and occasionally delightful) problems and exercises. • Abstract concepts are introduced gradually throughout the text. The major ideas are carefully developed at various levels of generality before the student is introduced to abstract vector spaces. • Discrete and continuous dynamical systems are used as a motivation for eigen vectors, and as a unifying theme thereafter. New Features in the Fourth Edition Students and instructors generally found the third edition to be accurate and well structured. While preserving the overall organization and character of the text, some changes seemed in order. • A large number of exercises have been added to the problem sets, from the elementary to the challenging. For example, two dozen new exercises on conic and cubic curve fitting lead up to a discussion of the Cramer-Euler Paradox on fitting a cubic through nine points. • The section on matrix products now precedes the discussion of the inverse of a matrix, making the presentation more sensible from an algebraic point of view. • Striking a balance between the earlier editions, the determinant is defined in terms of “patterns”, a transparent way to deal with permutations. Laplace expansion and Gaussian elimination are presented as alternative approaches. • There have been hundreds of small editorial improvements—offering a hint in a difficult problem for example—or choosing a more sensible notation in a theorem.

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Preface (with David Steinsaltz) police officer 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 Ato 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 five 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.” 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 identification of Euclidean space with R 3; the techniques and applications proliferated since his day. To organize and clarify those is the role of a modem 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 subjects of mathematics algorithms and calculations, or are they grand theories and abstrac- tions that evade the need for computation? If the former, is it important that the students learn to carry out the computations with pencil and paper, or should the al gorithm “press the calculator’s x ~ 1 button” be allowed to substitute for the traditional method of finding 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 . . . , ” and to appreciate in advance the possible outcomes of a particular computation. 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 o f 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 system’s behavior and understanding its dependence from initial conditions.

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xii Preface 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 sufficiently familiar with the inner workings of the algorithm. As the explicit compu tations are being replaced gradually by a theoretical overview of how the algorithm works, the burden of calculation will be taken up by technology, particularly for those wishing to carry out the more numerical and applied exercises. 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, chemistry, biology, economics, and, of course, mathematics itself. Often we have simplified 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 difficulty 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. 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 significant 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 specific 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 el\ 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. Outline of the Text Chapter I This chapter provides a careful introduction to the solution of systems of linear equations by Gauss-Jordan elimination. Once the concrete problem is solved, we restate it in terms of matrix formalism and discuss the geometric properties of the solutions. C hapter 2 Here we raise the abstraction a notch and reinterpret matrices as linear transformations. The reader is introduced to the modem notion of a function, as an arbitrary association between an input and an output, which leads into a dis cussion of inverses. The traditional method for finding the inverse of a matrix is explained: It fits in naturally as a sort of automated algorithm for Gauss-Jordan elimination.

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Preface xiii We define linear transformations primarily in terms of matrices, since that is how they are used; the abstract concept of linearity is presented as an auxiliary notion. Rotations, reflections, and orthogonal projections in R 2 are emphasized, both as archetypal, easily visualized examples, and as preparation for future applications. Chapter 3 We introduce the central concepts of linear algebra: subspaces, image and kernel, linear independence, bases, coordinates, and dimension, still firmly fixed in R". Chapter 4 Generalizing the ideas of the preceding chapter and using an abun dance of examples, we introduce abstract vector spaces (which are called linear spaces here, to prevent the confusion some students experience with the term “vector”). Chapter 5 This chapter includes some of the most basic applications of linear algebra to geometry and statistics. We introduce orthonormal bases and the Gram- Schmidt process, along with the QR factorization. The calculation of correlation coefficients is discussed, and the important technique of least-squares approxima tions is explained, in a number of different contexts. Chapter 6 Our discussion of determinants is algorithmic, based on the counting of patterns (a transparent way to deal with permutations). We derive the properties of the determinant from careful analysis of this procedure, tieing it together with Gauss- Jordan elimination. The goal is to prepare for the main application of determinants: the computation of characteristic polynomials. C hapter 7 This chapter introduces the central application of the latter half of the text: linear dynamical systems. We begin with discrete systems and are nat urally led to seek eigenvectors, which characterize the long-term behavior of the system. Qualitative behavior is emphasized, particularly stability conditions. Com plex eigenvalues are explained, without apology, and tied into earlier discussions of two-dimensional rotation matrices.

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xiv Preface Chapter 8 The ideas and methods of Chapter 7 are applied to geometry. We discuss the spectral theorem for symmetric matrices and its applications to quadratic forms, conic sections, and singular values. C hapter 9 Here we apply the methods developed for discrete dynamical systems to continuous ones, that is, to systems of first-order linear differential equations. Again, the cases of real and complex eigenvalues are discussed. Solutions Manuals • Student's Solutions Manual, with carefully worked solutions to all odd- numbered problems in the text (ISBN 0-13-600927-1) • Instructor's Solutions Manual, with solutions to all the problems in the text (ISBN 0-13-600928-X) Acknowledgments I first thank my students and colleagues at Harvard University, Colby College, and Koq University (Istanbul) for the key role they have played in developing this text out of a series of rough lecture notes. The following colleagues, who have taught the course with me, have made invaluable contributions: Attila A§kar Fernando Gouvea Barry Mazur Persi Diaconis Jan Holly David Mumford Jordan Ellenberg Varga Kalantarov David Steinsaltz Matthew Emerton David Kazhdan Shlomo Sternberg Edward Frenkel Oliver Knill Richard Taylor Alexandru Ghitza Leo Livshits George Welch I am grateful to those who have taught me algebra: Peter Gabriel, Volker Strassen, and Bartel van der Waerden at the University of Zurich; John Tate at Harvard University; Maurice Auslander at Brandeis University; and many more. I owe special thanks to John Boiler, William Calder, Devon Ducharme, and Robert Kaplan for their thoughtful review of the manuscript. I wish to thank Sylvie Bessette, Dennis Kletzing, and Laura Lawrie for the careful preparation of the manuscript and Paul Nguyen for his well-drawn figures. Special thanks go to Kyle Burke, who has been my assistant at Colby College for three years, having taken linear algebra with me as a first-year student. Paying close attention to the way his students understood or failed to understand the material, Kyle came up with new ways to explain many of the more troubling concepts (such as the kernel of a matrix or the notion of an isomorphism). Many of his ideas have found their way into this text. Kyle took the time to carefully review various iterations of the manuscript, and his suggestions and contributions have made this a much better text. I am grateful to those who have contributed to the book in many ways: Marilyn Baptiste, Menoo Cung, Srdjan Divac, Devon Ducharme, Robin Gottlieb, Luke Hunsberger, George Mani, Bridget Neale, Alec van Oot, Akilesh Palanisamy, Rita Pang, Esther Silberstein, Radhika de Silva, Jonathan Tannenhauser, Ken Wada- Shiozaki, Larry Wilson, and Jingjing Zhou.

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