Linear algebra and its applications

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
About the Author......Page 4
Contents......Page 6
Preface......Page 9
Acknowledgments......Page 15
A Note to Students......Page 16
 INTRODUCTORY EXAMPLE: Linear Models in Economics and Engineering......Page 18
 1.1 Systems of Linear Equations......Page 19
 1.2 Row Reduction and Echelon Forms......Page 29
 1.3 Vector Equations......Page 41
 1.4 The Matrix Equation Ax = b......Page 52
 1.5 Solution Sets of Linear Systems......Page 60
 1.6 Applications of Linear Systems......Page 67
 1.7 Linear Independence......Page 73
 1.8 Introduction to Linear Transformations......Page 80
 1.9 The Matrix of a Linear Transformation......Page 88
 1.10 Linear Models in Business, Science, and Engineering......Page 98
 Supplementary Exercises......Page 106
 INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design......Page 110
 2.1 Matrix Operations......Page 111
 2.2 The Inverse of a Matrix......Page 121
 2.3 Characterizations of Invertible Matrices......Page 130
 2.4 Partitioned Matrices......Page 136
 2.5 Matrix Factorizations......Page 142
 2.6 The Leontief Input–Output Model......Page 151
 2.7 Applications to Computer Graphics......Page 157
 2.8 Subspaces of Rn......Page 165
 2.9 Dimension and Rank......Page 172
 Supplementary Exercises......Page 179
 INTRODUCTORY EXAMPLE: Random Paths and Distortion......Page 182
 3.1 Introduction to Determinants......Page 183
 3.2 Properties of Determinants......Page 188
 3.3 Cramer’s Rule, Volume, and Linear Transformations......Page 196
 Supplementary Exercises......Page 205
 INTRODUCTORY EXAMPLE: Space Flight and Control Systems......Page 208
 4.1 Vector Spaces and Subspaces......Page 209
 4.2 Null Spaces, Column Spaces, and Linear Transformations......Page 217
 4.3 Linearly Independent Sets; Bases......Page 227
 4.4 Coordinate Systems......Page 235
 4.5 The Dimension of a Vector Space......Page 244
 4.6 Rank......Page 249
 4.7 Change of Basis......Page 258
 4.8 Applications to Difference Equations......Page 263
 4.9 Applications to Markov Chains......Page 272
 Supplementary Exercises......Page 281
 INTRODUCTORY EXAMPLE: Dynamical Systems and Spotted Owls......Page 284
 5.1 Eigenvectors and Eigenvalues......Page 285
 5.2 The Characteristic Equation......Page 293
 5.3 Diagonalization......Page 300
 5.4 Eigenvectors and Linear Transformations......Page 307
 5.5 Complex Eigenvalues......Page 314
 5.6 Discrete Dynamical Systems......Page 320
 5.7 Applications to Differential Equations......Page 330
 5.8 Iterative Estimates for Eigenvalues......Page 338
 Supplementary Exercises......Page 345
 INTRODUCTORY EXAMPLE: The North American Datum and GPS Navigation......Page 348
 6.1 Inner Product, Length, and Orthogonality......Page 349
 6.2 Orthogonal Sets......Page 357
 6.3 Orthogonal Projections......Page 366
 6.4 The Gram–Schmidt Process......Page 373
 6.5 Least-Squares Problems......Page 379
 6.6 Applications to Linear Models......Page 387
 6.7 Inner Product Spaces......Page 395
 6.8 Applications of Inner Product Spaces......Page 402
 Supplementary Exercises......Page 409
 INTRODUCTORY EXAMPLE: Multichannel Image Processing......Page 412
 7.1 Diagonalization of Symmetric Matrices......Page 414
 7.2 Quadratic Forms......Page 420
 7.3 Constrained Optimization......Page 427
 7.4 The Singular Value Decomposition......Page 433
 7.5 Applications to Image Processing and Statistics......Page 443
 Supplementary Exercises......Page 451
 INTRODUCTORY EXAMPLE: The Platonic Solids......Page 454
 8.1 Affine Combinations......Page 455
 8.2 Affine Independence......Page 463
 8.3 Convex Combinations......Page 473
 8.4 Hyperplanes......Page 480
 8.5 Polytopes......Page 488
 8.6 Curves and Surfaces......Page 500
 A Uniqueness of the Reduced Echelon Form......Page 512
 B Complex Numbers......Page 513
 C......Page 518
 E......Page 520
 I......Page 521
 L......Page 522
 O......Page 523
 P......Page 524
 R......Page 525
 S......Page 526
 Z......Page 527
Answers to Odd-Numbered Exercises......Page 528
 C......Page 568
 E......Page 569
 I......Page 570
 M......Page 571
 P......Page 572
 R......Page 573
 S......Page 574
 Z......Page 575
Photo Credits......Page 576