Undergraduate Texts in Mathematics Editors s. Axler F. w. Gehring K.A. Ribet Springer Science+Business Media, LLC
Undergraduate Texts in Mathematics Abbott: Understanding Analysis. Anglin: Mathematics: A Concise History and Philosophy. Readings in Mathematics. Anglin/Lambek: The Heritage of Thales. Readings in Mathematics. Apostol: Introduction to Analytic Number Theory. Second edition. Armstrong: Basic Topology. Armstrong: Groups and Symmetry. Axler: Linear Algebra Done Right. Second edition. Beardon: Limits: A New Approach to Real Analysis. BaklNewman: Complex Analysis. Second edition. BanchofflWermer: Linear Algebra Through Geometry. Second edition. Berberian: A First Course in Real Analysis. Bix: Conics and Cubics: A Concrete Introduction to Algebraic Curves. Bremaud: An Introduction to Probabilistic Modeling. Bressoud: Factorization and Primality Testing. Bressoud: Second Year Calculus. Readings in Mathematics. Brickman: Mathematical Introduction to Linear Programming and Game Theory. Browder: Mathematical Analysis: An Introduction. Buchmann: Introduction to Cryptography. Buskes/van Rooij: Topological Spaces: From Distance to Neighborhood. Callahan: The Geometry of Spacetime: An Introduction to Special and General Relavitity. Carter/van Brunt: The Lebesgue Stieltjes Integral: A Practical Introduction. Childs: A Concrete Introduction to Higher Algebra. Second edition. Chung/AitSahlia: Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance. Fourth edition. CoxlLittle/O'Shea: Ideals, Varieties, and Algorithms. Second edition. Croom: Basic Concepts of Algebraic Topology. Curtis: Linear Algebra: An Introductory Approach. Fourth edition. Devlin: The Joy ofsets: Fundamentals of Contemporary Set Theory. Second edition. Dixmier: General Topology. Driver: Why Math? Ebbinghaus/FlumIThomas: Mathematical Logic. Second edition. Edgar: Measure, Topology, and Fractal Geometry. Elaydi: An Introduction to Difference Equations. Second edition. Erdos/Suninyi: Topics in the Theory of Numbers. Estep: Practical Analysis in One Variable. Exner: An Accompaniment to Higher Mathematics. Exner: Inside Calculus. FinelRosenberger: The Fundamental Theory of Algebra. Fischer: Intermediate Real Analysis. Flanigan/Kazdan: Calculus Two: Linear and Nonlinear Functions. Second edition. Fleming: Functions of Several Variables. Second edition. Foulds: Combinatorial Optimization for Undergraduates. Foulds: Optimization Techniques: An Introduction. Franklin: Methods ofmathematical Economics. Frazier: An Introduction to Wavelets Through Linear Algebra.
Charles W. Curtis Linear Algebra An Introductory Approach With 37 Illustrations, Springer
Charles W. Curtis Department of Mathematics University of Oregon Eugene, OR 97403 USA Editorial Board S. Axler Department of Mathematics San Francisco State University San Francisco, CA 94132 USA EW. Gehring Department of Mathematics East Hali University of Michigan Ann Arbor, MI 48109 USA K.A. Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 15-01 Library of Congress Cataloging-in-Publication Data Curtis, Charles W. Linear algebra. (Undergraduate texts in mathematics) Bibliography: p. lncludes index. 1. Algebras, Linear. 1. Title. II. Series. QA184.C87 1984 515'.55 84-1408 Printed on acid-free paper. Previous editions of this book were published by Allyn and Bacon, Inc. : Boston. 1974, 1984 Springer Science+Business Media New York Originally published by Springer-Verlag New York.lnc. in 1984 Softcover reprint of the hardcover 4th edition 1984 AII rights reserved. This work may not be translated or copied in whole or in pan without the written permission of the publisher Springer Science+Business Media. LLC except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and relrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, Irademarks. etc., in this publication, even if the former are not especially identified, is not 10 be taken as a sign that such names, as understood by thc Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 978-1-4612-7019-5 ISBN 978-1-4612-1136-5 (ebook) DOI 10.1007/978-1-4612-1136-5
To AMY, SAM, AND SPENCER
Preface Linear algebra is the branch of mathematics that has grown from a careful study of the problem of solving systems of linear equations. The ideas that developed in this way have become part of the language of much of higher mathematics. They also provide a framework for applications of linear algebra to many problems in mathematics, the natural sciences, economics, and computer science. This book is the revised fourth edition of a textbook designed for upper division courses in linear algebra. While it does not presuppose an earlier course, many connections between linear algebra and undergraduate analysis are worked into the discussion, making it best suited for students who have completed the calculus sequence. For many students, this may be the first course in which proofs of the main results are presented on an equal footing with methods for solving numerical problems. The concepts needed to understand the proofs are shown to emerge naturally from attempts to solve concrete problems. This connection is illustrated by worked examples in almost every section. Many numerical exercises are included, which use all the ideas, and develop important techniques for problem-solving. There are also theoretical exercises, which provide opportunities for students to discover interesting things for themselves, and to write mathematical explanations in a convincing way. Answers and hints for many of the problems are given in the back. Not all answers are given, however, to encourage students to learn how to check their work. A special feature of the book is the inclusion of sections devoted to applications of linear algebra. These are: IO on the geometric interpretation of systems of linear equations, 14 and 33 on finite symmetry groups in two and three dimensions, 34 on systems of first-order differential equations and the exponential of a matrix, and 36 on the composition of quadratic forms. This edition contains a new section ( 35) on analytic methods in matrix theory, with applications to Markov chains vii
viii PREFACE in probability theory. These sections are not required in the mainstream of the discussion, and can either be put in a course, or used for independent study. Another feature of the book is that it contains an introduction to the axiomatic methods of modem algebra. The basic systems of algebragroups, rings, and fields-all occur naturally in linear algebra. Their definitions and elementary properties are discussed as they come up in the text. This book can be used as a text for either an elementary or an advanced course. An elementary course of one semester or two quarters can be based on Chapters 2-6, 22-24 from Chapter 7, and 30, 31 from Chapter 9. In a first course, it would be reasonable to take a less abstract approach, considering only vector spaces over the field of real numbers, and pointing out that the main theorems also hold for vector spaces over the field of complex numbers, before starting Chapter 7. Some introductory remarks about the field of real numbers and mathematical induction, from 2, would be needed at the beginning. An advanced undergraduate course, for students who have had an elementary course, can be based on Chapters 6-9, along with as much of Chapter 10 as time permits. It is a pleasure to acknowledge the encouragement I have received for this project from students in my classes at Wisconsin-Madison and Oregon. Their comments, and suggestions from instructors who taught from previous editions, have led to improvements from one edition to the next. I am also grateful for the interest and support from my family. Eugene, Oregon April 16, 1996 Charles W. Curtis
Contents 1. Introduction to Linear Algebra 1 1. Some problems which lead to linear algebra 1 2. Number systems and mathematical induction 6 2. Vector Spaces and Systems of Linear Equations 16 3. Vector spaces 16 4. Subspaces and linear dependence 26 5. The concepts of basis and dimension 34 6. Row equivalence of matrices 38 7. Some general theorems about finitely generated vector spaces 48 8. Systems of linear equations 53 9. Systems of homogeneous equations 62 10. Linear manifolds 69 3. Linear Transformations and Matrices II. Linear transformations 75 12. Addition and multiplication of matrices 13. Linear transformations and matrices 4. Vector Spaces with an Inner Product 14. The concept of symmetry 109 15. Inner products 119 ix 99 88 75 109
x CONTENTS s. Determinants 16. Definition of determinants 132 17. Existence and uniqueness of determinants 18. The multiplication theorem for determinants 19. Further properties of determinants 150 6. Polynomials and Complex Numbers 20. Polynomials 163 21. Complex numbers 176 7. The Theory of a Single Linear Transformation 22. Basic concepts 184 23. Invariant subspaces 193 24. The triangular form theorem 201 25. The rational and Jordan canonical forms 140 146 216 132 163 184 8. Dual Vector Spaces and Multilinear Algebra 26. Quotient spaces and dual vector spaces 228 27. Bilinear forms and duality 236 28. Direct sums and tensor products 243 29. A proof of the elementary divisor theorem 260 228 9. Orthogonal and Unitary Transformations 266 30. The structure of orthogonal transformations 266 31. The principal axis theorem 271 32. Unitary transformations and the spectral theorem 278 10. Some Applications of Linear Algebra 289 33. Finite symmetry groups in three dimensions 289 34. Application to differential equations 298 35. Analytic methods in matrix theory 306 36. Sums of squares and Hurwitz's theorem 315 Bibliography (with Notes) 325 Solutions of Selected Exercises 327 Symbols (Including Greek Letters) 342 Index 344