Chapter 1 Linear Equations in Linear Algebra 1
INTRODUCTORY EXAMPLE: Linear Models in Economics and Engineering 1
1.1 Systems of Linear Equations 2
1.2 Row Reduction and Echelon Forms 14
1.3 Vector Equations 28
1.4 The Matrix Equation Ax =b 40
1.5 Solution Sets of Linear Systems 50
1.6 Applications of Linear Systems 57
1.7 Linear Independence 65
1.8 Introduction to Linear Transformations 73
1.9 The Matrix of a Linear Transformation 82
1.10 Linear Models in Business, Science, and Engineering 92
Supplementary Exercises. 102
Chapter 2 Matrix Alflebra 105
INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design 105
2.1 Matrix Operations 107
2.2 The Inverse of a Matrix 118
2.3 Characterizations of Invertible Matrices 128
2.4 Partitioned Matrices 134
2.5 Matrix Factorizations 142
2.6 The Leontief Input-Output Model 152
2.7 Applications to Computer Graphics 158
2.8 Subspaces of R 167
2.9 Dimension and Rank 176
Supplementary Exercises 183
Chapter 3 Determinants 185
INTRODUCTORY EXAMPLE: Determinants in Analytic Geometry 185
3.1 Introduction to Determinants 186
3.2 Properties of Determinants 192
3.3 Cramer's Rule, Volume, and Linear Transformations 201
Supplementary Exercises 211
Chapter 4 Vector Spaces 215
INTRODUCTORY EXAMPLE: Space Flight and Control Systems 215
4.1 Vector Spaces and Subspaces 216
4.2 Null Spaces, Column Spaces, and Linear Transformations 226
4.3 Linearly Independent Sets; Bases 237
4.4 Coordinate Systems 246
4.5 The Dimension of a Vector Space 256
4.6 Rank 262
4.7 Change of Basis 271
4.8 Applications to Difference Equations 277
4.9 Applications to Markov Chains 288
Supplementary Exercises 299
Chapter 5 Eigenvalues and Eigenvectors 301
INTRODUCTORY EXAMPLE' Dynamical Systems and Spotted Owls 301
5.1 Eigenvectors and Eigenvalues 302
512 The Characteristic Equation 310
5.3 Diagonalization 319
5.4 Eigenvectors and Linear Transformations 327
5.5 Complex Eigenvaiues 335
5.6 Discrete Dynamical Systems 342
5.7 Applications to Differential Equations 353
5.8 terative Estimates for Eigenvalues 363
Supplementary Exercises 370
Chapter 6 Orthogonality and Least Squares 373
INSTRODUCTORY EXAMPLE: Readjusting the North American Datum 373
6.1 Inner Product, Length, and Orthogonality 375
6.2 Orthogonal Sets 384
6.3 Orthogonal Projections 394
6.4 The Gram-Schmidt Process 402
6.5 Least-Squares Problems 409
6.6 Applications to Linear Models 419
6.7 Inner Product Spaces 427
6.8 Applications of Inner Product Spaces 436
Supplementary Exercises 444
Chapter 7 Symmetric Matrices and Quadratic Forms 447
INSTRODUCTORY EXAMPLE: Multichannel Image Processing 447
7.1 Diagonalization of Symmetric Matrices 449
7.2 Quadratic Forms 455
7.3 Constrained Optimization 463
7.4 The Singular Value Decomposition 471
7.5 Applications to Image Processing and Statistics 482
Supplementary Exercises 491
Appendixes
A Uniqueness of the Reduced Echelon Form A1
B Complex Numbers A3
Glossary A9
Answers to Odd-Numbered Exercises A19
Index II
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