Graduate Texts in Mathematics 135

Transcription

1 Graduate Texts in Mathematics 135 S. Axler Editorial Board F.W. Gehring K.A. Ribet

2 Graduate Texts in Mathematics 1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 3 SCHAEFER. Topological Vector Spaces. 2nd ed. 4 HILTON/STAMMBACH. A Course in Homological Algebra. 2nd ed. 5 MAC LANE. Categories for the Working Mathematician. 2nd ed. 6 HUGHES/PIPER. Projective Planes. 7 J.-P. SERRE. A Course in Arithmetic. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory. 10 COHEN. A Course in Simple Homotopy Theory. 11 CONWAY. Functions of One Complex Variable I. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 ANDERSON/FULLER. Rings and Categories of Modules. 2nd ed. 14 GOLUBITSKY/GUILLEMIN. Stable Mappings and Their Singularities. 15 BERBERIAN. Lectures in Functional Analysis and Operator Theory. 16 WrNTER. The Structure of Fields. 17 ROSENBLATT. Random Processes. 2nd ed. 18 HALMOS. Measure Theory. 19 HALMOS. A Hilbert Space Problem Book. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 21 HUMPHREYS. Linear Algebraic Groups. 22 BARNES/]VIACK. An Algebraic Introduction to Mathematical Logic. 23 GREUB. Linear Algebra. 4th ed. 24 HOLMES. Geometric Functional Analysis and Its Applications. 25 HEWITT/STROMBERG. Real and Abstract Analysis. 26 MANES. Algebraic Theories. 27 KELLEY. General Topology. 28 ZARISKI/SAMUEL. Commutative Algebra. Vol.I. 29 ZAPdSrO/SAMUEL. Commutative Algebra. Vol.II. 30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts. 31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra. 32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory. 33 HIRSCH. Differential Topology. 34 SPITZER. Principles of Random Walk. 2nd ed. 35 ALEXANDE~ERMER. Several Complex Variables and Banach Algebras. 3rd ed. 36 KELLEY/NAMIOKA et al. Linear Topological Spaces. 37 MONK. Mathematical Logic. 38 GRAUERT/FRITZSCHE. Several Complex Variables. 39 ARVESON. An Invitation to C*-Algebras. 40 KEMENY/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed. 41 APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 2nd ed. 42 J.-P. SERRE. Linear Representations of Finite Groups. 43 GILLMAN/JERISON. Rings of Continuous Functions. 44 KENDIG. Elementary Algebraic Geometry. 45 LOEVE. Probability Theory I. 4th ed. 46 LoI~vE. Probability Theory II. 4th ed. 47 MOISE. Geometric Topology in Dimensions 2 and SACHS/~qVu. General Relativity for Mathematicians. 49 GRUENBERG/~vVEIR. Linear Geometry. 2nd ed. 50 EDWARDS. Fermat's Last Theorem. 51 KLINGENBERG. A Course in Differential Geometry. 52 HARTSHORNE. Algebraic Geometry. 53 MANN. A Course in Mathematical Logic. 54 GRAVEK/V~ATKINS. Combinatorics with Emphasis on the Theory of Graphs. 55 BROWN/PEARCY. Introduction to Operator Theory I: Elements of Functional Analysis. 56 MASSEY. Algebraic Topology: An Introduction. 57 CROWELL/Fox. Introduction to Knot Theory. 58 KOBLITZ. p-adic Numbers, p-adic Analysis, and Zeta-Functions. 2nd ed. 59 LANG. Cyclotomic Fields. 60 ARNOLD. Mathematical Methods in Classical Mechanics. 2nd ed. 61 WHITEHEAD. Elements of Homotopy Theory. 62 KARGAPOLOV/]~ERLZJAKOV. Fundamentals of the Theory of Groups. 63 BOLLOBAS. Graph Theory. (continued after index)

3 Steven Roman Advanced Linear Algebra Second Edition Springer

4 Steven Roman University of California, Irvine Irvine, California USA Editorial Board." S. Axler EW. Gehring K.A. Ribet Mathematics Department Mathematics Department Mathematics Department San Francisco State East Hall University of California, University University of Michigan Berkeley San Francisco, CA Ann Arbor, MI Berkeley, CA USA USA USA Mathematics Subject Classification (2000): 15-xx Library of Congress Cataloging-in-Publication Data Roman, Steven. Advanced linear algebra / Steven Roman.--2nd ed. p. cm. Includes bibliographical references and index. ISBN (acid-free paper) 1. Algebras, Linear. I. Title. QA184.2.R '.5-- dc ISBN Printed on acid-free paper Steven Roman All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+ Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America springer.com

5 To Donna and to my poker buddies Rachelle, Carol and Dan

6 Preface to the Second Edition Let me begin by thanking the readers of the first edition for their many helpful comments and suggestions. The second edition represents a major change from the first edition. Indeed, one might say that it is a totally new book, with the exception of the general range of topics covered. The text has been completely rewritten. I hope that an additional 12 years and roughly 20 books worth of experience has enabled me to improve the quality of my exposition. Also, the exercise sets have been completely rewritten. The second edition contains two new chapters: a chapter on convexity, separation and positive solutions to linear systems (Chapter 15) and a chapter on the QR decomposition, singular values and pseudoinverses (Chapter 17). The treatments of tensor products and the umbral calculus have been greatly expanded and I have included discussions of determinants (in the chapter on tensor products), the complexification of a real vector space, Schur's lemma and Geršgorin disks. Steven Roman Irvine, California February 2005

7 Preface to the First Edition This book is a thorough introduction to linear algebra, for the graduate or advanced undergraduate student. Prerequisites are limited to a knowledge of the basic properties of matrices and determinants. However, since we cover the basics of vector spaces and linear transformations rather rapidly, a prior course in linear algebra (even at the sophomore level), along with a certain measure of mathematical maturity, is highly desirable. Chapter 0 contains a summary of certain topics in modern algebra that are required for the sequel. This chapter should be skimmed quickly and then used primarily as a reference. Chapters 1 3 contain a discussion of the basic properties of vector spaces and linear transformations. Chapter 4 is devoted to a discussion of modules, emphasizing a comparison between the properties of modules and those of vector spaces. Chapter 5 provides more on modules. The main goals of this chapter are to prove that any two bases of a free module have the same cardinality and to introduce noetherian modules. However, the instructor may simply skim over this chapter, omitting all proofs. Chapter 6 is devoted to the theory of modules over a principal ideal domain, establishing the cyclic decomposition theorem for finitely generated modules. This theorem is the key to the structure theorems for finite-dimensional linear operators, discussed in Chapters 7 and 8. Chapter 9 is devoted to real and complex inner product spaces. The emphasis here is on the finite-dimensional case, in order to arrive as quickly as possible at the finite-dimensional spectral theorem for normal operators, in Chapter 10. However, we have endeavored to state as many results as is convenient for vector spaces of arbitrary dimension. The second part of the book consists of a collection of independent topics, with the one exception that Chapter 13 requires Chapter 12. Chapter 11 is on metric vector spaces, where we describe the structure of symplectic and orthogonal geometries over various base fields. Chapter 12 contains enough material on metric spaces to allow a unified treatment of topological issues for the basic

8 x Preface Hilbert space theory of Chapter 13. The rather lengthy proof that every metric space can be embedded in its completion may be omitted. Chapter 14 contains a brief introduction to tensor products. In order to motivate the universal property of tensor products, without getting too involved in categorical terminology, we first treat both free vector spaces and the familiar direct sum, in a universal way. Chapter 15 [Chapter 16 in the second edition] is on affine geometry, emphasizing algebraic, rather than geometric, concepts. The final chapter provides an introduction to a relatively new subject, called the umbral calculus. This is an algebraic theory used to study certain types of polynomial functions that play an important role in applied mathematics. We give only a brief introduction to the subject emphasizing the algebraic aspects, rather than the applications. This is the first time that this subject has appeared in a true textbook. One final comment. Unless otherwise mentioned, omission of a proof in the text is a tacit suggestion that the reader attempt to supply one. Steven Roman Irvine, California

9 Contents Preface to the Second Edition, vii Preface to the First Edition, ix Preliminaries, 1 Part 1 Preliminaries, 1 Part 2 Algebraic Structures, 16 Part I Basic Linear Algebra, 31 1 Vector Spaces, 33 Vector Spaces, 33 Subspaces, 35 Direct Sums, 38 Spanning Sets and Linear Independence, 41 The Dimension of a Vector Space, 44 Ordered Bases and Coordinate Matrices, 47 The Row and Column Spaces of a Matrix, 48 The Complexification of a Real Vector Space, 49 Exercises, 51 2 Linear Transformations, 55 Linear Transformations, 55 Isomorphisms, 57 The Kernel and Image of a Linear Transformation, 57 Linear Transformations from to, 59 The Rank Plus Nullity Theorem, 59 Change of Basis Matrices, 60 The Matrix of a Linear Transformation, 61 Change of Bases for Linear Transformations, 63 Equivalence of Matrices, 64 Similarity of Matrices, 65 Similarity of Operators, 66 Invariant Subspaces and Reducing Pairs, 68

10 xii Contents Topological Vector Spaces, 68 Linear Operators on, 71 Exercises, 72 3 The Isomorphism Theorems, 75 Quotient Spaces, 75 The Universal Property of Quotients and the First Isomorphism Theorem, 77 Quotient Spaces, Complements and Codimension, 79 Additional Isomorphism Theorems, 80 Linear Functionals, 82 Dual Bases, 83 Reflexivity, 84 Annihilators, 86 Operator Adjoints, 88 Exercises, 90 4 Modules I: Basic Properties, 93 Modules, 93 Motivation, 93 Submodules, 95 Spanning Sets, 96 Linear Independence, 98 Torsion Elements, 99 Annihilators, 99 Free Modules, 99 Homomorphisms, 100 Quotient Modules, 101 The Correspondence and Isomorphism Theorems, 102 Direct Sums and Direct Summands, 102 Modules Are Not As Nice As Vector Spaces, 106 Exercises, Modules II: Free and Noetherian Modules, 109 The Rank of a Free Module, 109 Free Modules and Epimorphisms, 114 Noetherian Modules, 115 The Hilbert Basis Theorem, 118 Exercises, Modules over a Principal Ideal Domain, 121 Annihilators and Orders, 121 Cyclic Modules, 122 Free Modules over a Principal Ideal Domain, 123 Torsion-Free and Free Modules, 125

11 Contents xiii Prelude to Decomposition: Cyclic Modules, 126 The First Decomposition, 127 A Look Ahead, 127 The Primary Decomposition, 128 The Cyclic Decomposition of a Primary Module, 130 The Primary Cyclic Decomposition Theorem, 134 The Invariant Factor Decomposition, 135 Exercises, The Structure of a Linear Operator, 141 A Brief Review, 141 The Module Associated with a Linear Operator, 142 Orders and the Minimal Polynomial, 144 Cyclic Submodules and Cyclic Subspaces, 145 Summary, 147 The Decomposition of, 147 The Rational Canonical Form, 148 Exercises, Eigenvalues and Eigenvectors, 153 The Characteristic Polynomial of an Operator, 153 Eigenvalues and Eigenvectors, 155 Geometric and Algebraic Multiplicities, 157 The Jordan Canonical Form, 158 Triangularizability and Schur's Lemma, 160 Diagonalizable Operators, 165 Projections, 166 The Algebra of Projections, 167 Resolutions of the Identity, 170 Spectral Resolutions, 172 Projections and Invariance, 173 Exercises, Real and Complex Inner Product Spaces, 181 Norm and Distance, 183 Isometries, 186 Orthogonality, 187 Orthogonal and Orthonormal Sets, 188 The Projection Theorem and Best Approximations, 192 Orthogonal Direct Sums, 194 The Riesz Representation Theorem, 195 Exercises, Structure Theory for Normal Operators, 201 The Adjoint of a Linear Operator, 201

12 xiv Contents Unitary Diagonalizability, 204 Normal Operators, 205 Special Types of Normal Operators, 207 Self-Adjoint Operators, 208 Unitary Operators and Isometries, 210 The Structure of Normal Operators, 215 Matrix Versions, 222 Orthogonal Projections, 223 Orthogonal Resolutions of the Identity, 226 The Spectral Theorem, 227 Spectral Resolutions and Functional Calculus, 228 Positive Operators, 230 The Polar Decomposition of an Operator, 232 Exercises, 234 Part II Topics, Metric Vector Spaces: The Theory of Bilinear Forms, 239 Symmetric, Skew-Symmetric and Alternate Forms, 239 The Matrix of a Bilinear Form, 242 Quadratic Forms, 244 Orthogonality, 245 Linear Functionals, 248 Orthogonal Complements and Orthogonal Direct Sums, 249 Isometries, 252 Hyperbolic Spaces, 253 Nonsingular Completions of a Subspace, 254 The Witt Theorems: A Preview, 256 The Classification Problem for Metric Vector Spaces, 257 Symplectic Geometry, 258 The Structure of Orthogonal Geometries: Orthogonal Bases, 264 The Classification of Orthogonal Geometries: Canonical Forms, 266 The Orthogonal Group, 272 The Witt's Theorems for Orthogonal Geometries, 275 Maximal Hyperbolic Subspaces of an Orthogonal Geometry, 277 Exercises, Metric Spaces, 283 The Definition, 283 Open and Closed Sets, 286 Convergence in a Metric Space, 287 The Closure of a Set, 288

13 Contents xv Dense Subsets, 290 Continuity, 292 Completeness, 293 Isometries, 297 The Completion of a Metric Space, 298 Exercises, Hilbert Spaces, 307 A Brief Review, 307 Hilbert Spaces, 308 Infinite Series, 312 An Approximation Problem, 313 Hilbert Bases, 317 Fourier Expansions, 318 A Characterization of Hilbert Bases, 328 Hilbert Dimension, 328 A Characterization of Hilbert Spaces, 329 The Riesz Representation Theorem, 331 Exercises, Tensor Products, 337 Universality, 337 Bilinear Maps, 341 Tensor Products, 343 When Is a Tensor Product Zero? 348 Coordinate Matrices and Rank, 350 Characterizing Vectors in a Tensor Product, 354 Defining Linear Transformations on a Tensor Product, 355 The Tensor Product of Linear Transformations, 357 Change of Base Field, 359 Multilinear Maps and Iterated Tensor Products, 363 Tensor Spaces, 366 Special Multilinear Maps, 371 Graded Algebras, 372 The Symmetric Tensor Algebra, 374 The Antisymmetric Tensor Algebra: The Exterior Product Space, 380 The Determinant, 387 Exercises, Positive Solutions to Linear Systems: Convexity and Separation 395 Convex, Closed and Compact Sets, 398 Convex Hulls, 399

14 xvi Contents Linear and Affine Hyperplanes, 400 Separation, 402 Exercises, Affine Geometry, 409 Affine Geometry, 409 Affine Combinations, 41 Affine Hulls, 412 The Lattice of Flats, 413 Affine Independence, 416 Affine Transformations, 417 Projective Geometry, 419 Exercises, Operator Factorizations: QR and Singular Value, 425 The QR Decomposition, 425 Singular Values, 428 The Moore Penrose Generalized Inverse, 430 Least Squares Approximation, 433 Exercises, The Umbral Calculus, 437 Formal Power Series, 437 The Umbral Algebra, 439 Formal Power Series as Linear Operators, 443 Sheffer Sequences, 446 Examples of Sheffer Sequences, 454 Umbral Operators and Umbral Shifts, 456 Continuous Operators on the Umbral Algebra, 458 Operator Adjoints, 459 Umbral Operators and Automorphisms of the Umbral Algebra, 460 Umbral Shifts and Derivations of the Umbral Algebra, 465 The Transfer Formulas, 470 A Final Remark, 471 Exercises, 472 References, 473 Index, 475