Elementary Linear Algebra, Applications Version

Cover......Page 1
Title Page......Page 5
Copyright Page......Page 6
Dedication......Page 7
Preface......Page 8
CONTENTS......Page 12
CHAPTER 1 Systems of Linear Equations and Matrices......Page 15
1.1 Introduction to Systems of Linear Equations......Page 16
1.2 Gaussian Elimination......Page 25
1.3 Matrices and Matrix Operations......Page 39
1.4 Inverses; Algebraic Properties of Matrices......Page 53
1.5 Elementary Matrices and a Method for Finding A-1......Page 66
1.6 More on Linear Systems and Invertible Matrices......Page 75
1.7 Diagonal, Triangular, and Symmetric Matrices......Page 81
1.8 Matrix Transformations......Page 89
Network Analysis (Traffic Flow)......Page 98
Electrical Circuits......Page 100
Balancing Chemical Equations......Page 102
Polynomial Interpolation......Page 105
1.10 Application: Leontief Input-Output Models......Page 110
2.1 Determinants by Cofactor Expansion......Page 119
2.2 Evaluating Determinants by Row Reduction......Page 127
2.3 Properties of Determinants; Cramer’s Rule......Page 132
3.1 Vectors in 2-Space, 3-Space, and n-Space......Page 145
3.2 Norm, Dot Product, and Distance in Rn......Page 156
3.3 Orthogonality......Page 169
3.4 The Geometry of Linear Systems......Page 178
3.5 Cross Product......Page 186
4.1 Real Vector Spaces......Page 197
4.2 Subspaces......Page 205
4.3 Linear Independence......Page 216
4.4 Coordinates and Basis......Page 226
4.5 Dimension......Page 235
4.6 Change of Basis......Page 243
4.7 Row Space, Column Space, and Null Space......Page 251
4.8 Rank, Nullity, and the Fundamental Matrix Spaces......Page 262
4.9 Basic Matrix Transformations in R2 and R3......Page 273
4.10 Properties of Matrix Transformations......Page 284
4.11 Application: Geometry of Matrix Operators on R2......Page 294
5.1 Eigenvalues and Eigenvectors......Page 305
5.2 Diagonalization......Page 316
5.3 Complex Vector Spaces......Page 327
5.4 Application: Differential Equations......Page 340
5.5 Application: Dynamical Systems and Markov Chains......Page 346
6.1 Inner Products......Page 359
6.2 Angle and Orthogonality in Inner Product Spaces......Page 369
6.3 Gram–Schmidt Process; QR-Decomposition......Page 378
6.4 Best Approximation; Least Squares......Page 392
6.5 Application: Mathematical Modeling Using Least Squares......Page 401
6.6 Application: Function Approximation; Fourier Series......Page 408
7.1 Orthogonal Matrices......Page 415
7.2 Orthogonal Diagonalization......Page 423
7.3 Quadratic Forms......Page 431
7.4 Optimization Using Quadratic Forms......Page 443
7.5 Hermitian, Unitary, and Normal Matrices......Page 451
8.1 General Linear Transformations......Page 461
8.2 Compositions and Inverse Transformations......Page 472
8.3 Isomorphism......Page 480
8.4 Matrices for General Linear Transformations......Page 486
8.5 Similarity......Page 495
9.1 LU-Decompositions......Page 505
9.2 The Power Method......Page 515
9.3 Comparison of Procedures for Solving Linear Systems......Page 523
9.4 Singular Value Decomposition......Page 528
9.5 Application: Data Compression Using Singular Value Decomposition......Page 535
CHAPTER 10 Applications of Linear Algebra......Page 541
10.1 Constructing Curves and Surfaces Through Specified Points......Page 542
10.2 The Earliest Applications of Linear Algebra......Page 547
10.3 Cubic Spline Interpolation......Page 554
10.4 Markov Chains......Page 565
10.5 Graph Theory......Page 575
10.6 Games of Strategy......Page 584
10.7 Leontief Economic Models......Page 593
10.8 Forest Management......Page 602
10.9 Computer Graphics......Page 609
10.10 Equilibrium Temperature Distributions......Page 617
10.11 Computed Tomography......Page 627
10.12 Fractals......Page 638
10.13 Chaos......Page 653
10.14 Cryptography......Page 666
10.15 Genetics......Page 677
10.16 Age-Specific Population Growth......Page 687
10.17 Harvesting of Animal Populations......Page 697
10.18 A Least Squares Model for Human Hearing......Page 705
10.19 Warps and Morphs......Page 711
10.20 Internet Search Engines......Page 720
APPENDIX A Working with Proofs......Page 729
APPENDIX B Complex Numbers......Page 733
Answers to Exercises......Page 741
Index......Page 787
Index of Applications and Historical Topics......Page 801