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Differential Equations and Linear Algebra 4th Stephen W. Goode
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FOURTH EDITION
Differential Equations
and Linear Algebra
Stephen W. Goode
and
Scott A. Annin
California State University, Fullerton
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Library of Congress Cataloging-in-Publication Data
Goode, Stephen W.,
Differential equations and linear algebra / Stephen W. Goode and Scott A. Annin,
California State University, Fullerton. — 4th edition.
pages cm
Includes index
ISBN 978-0-321-96467-0 — ISBN 0-321-96467-5
1. Differential equations. 2. Algebras, Linear. I. Annin, Scott. II. Title.
QA371.G644 2015
515’.35—dc23 2014006015
1 2 3 4 5 6 7 8 9 10—V031—19 18 17 16 15
www.pearsonhighered.com
ISBN 10: 0-321-96467-5
ISBN 13: 978-0-321-96467-0

Contents
Preface vii
1 First-Order Differential Equations 1
1.1 Differential Equations Everywhere 1
1.2 Basic Ideas and Terminology 13
1.3 The Geometry of First-Order Differential Equations 23
1.4 Separable Differential Equations 34
1.5 Some Simple Population Models 45
1.6 First-Order Linear Differential Equations 53
1.7 Modeling Problems Using First-Order Linear
Differential Equations 61
1.8 Change of Variables 71
1.9 Exact Differential Equations 82
1.10 Numerical Solution to First-Order Differential
Equations 93
1.11 Some Higher-Order Differential Equations 101
1.12 Chapter Review 106
2 Matrices and Systems of Linear
Equations 114
2.1 Matrices: Denitions and Notation 115
2.2 Matrix Algebra 122
2.3 Terminology for Systems of Linear Equations 138
2.4 Row-Echelon Matrices and Elementary Row
Operations 146
2.5 Gaussian Elimination 156
2.6 The Inverse of a Square Matrix 168
2.7 Elementary Matrices and the LU Factorization 179
2.8 The Invertible Matrix Theorem I 188
2.9 Chapter Review 190
3Determinants 196
3.1 The Denition of the Determinant 196
3.2 Properties of Determinants 209
3.3 Cofactor Expansions 222
3.4 Summary of Determinants 235
3.5 Chapter Review 242
iii

iv Contents
4VectorSpaces 246
4.1 Vectors in R
n
248
4.2 Denition of a Vector Space 252
4.3 Subspaces 263
4.4 Spanning Sets 274
4.5 Linear Dependence and Linear Independence 284
4.6 Bases and Dimension 298
4.7 Change of Basis 311
4.8 Row Space and Column Space 319
4.9 The Rank-Nullity Theorem 325
4.10 Invertible Matrix Theorem II 331
4.11 Chapter Review 332
5InnerProductSpaces 339
5.1 Denition of an Inner Product Space 340
5.2 Orthogonal Sets of Vectors and Orthogonal
Projections 352
5.3 The Gram-Schmidt Process 362
5.4 Least Squares Approximation 366
5.5 Chapter Review 376
6 Linear Transformations 379
6.1 Denition of a Linear Transformation 380
6.2 Transformations of R
2
391
6.3 The Kernel and Range of a Linear Transformation 397
6.4 Additional Properties of Linear Transformations 407
6.5 The Matrix of a Linear Transformation 419
6.6 Chapter Review 428
7EigenvaluesandEigenvectors 433
7.1 The Eigenvalue/Eigenvector Problem 434
7.2 General Results for Eigenvalues and Eigenvectors 446
7.3 Diagonalization 454
7.4 An Introduction to the Matrix Exponential Function 462
7.5 Orthogonal Diagonalization and Quadratic Forms 466
7.6 Jordan Canonical Forms 475
7.7 Chapter Review 488
8 Linear Differential Equations of
Order n 493
8.1 General Theory for Linear Differential Equations 495
8.2 Constant Coefcient Homogeneous Linear
Differential Equations 505
8.3 The Method of Undetermined Coefcients:
Annihilators 515
8.4 Complex-Valued Trial Solutions 526
8.5 Oscillations of a Mechanical System 529
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