CHAPTER 1 Systems of Linear Equations and Matrices 1
1.1 Introduction to Systems of Linear Equations 2
1.2 Gaussian Elimination 11
1.3 Matrices and Matrix Operations 25
1.4 Inverses; Algebraic Properties of Matrices 38
1.5 Elementary Matrices and a Method for Finding A?1 51
1.6 More on Linear Systems and Invertible Matrices 60
1.7 Diagonal, Triangular, and Symmetric Matrices 66
1.8 Application: Applications of Linear Systems 73
1.9 Application: Leontief Input-Output Models 85
CHAPTER 2 Determinants 93
2.1 Determinants by Cofactor Expansion 93
2.2 Evaluating Determinants by Row Reduction 100
2.3 Properties of Determinants; Cramer's Rule 106
CHAPTER 3 Euclidean Vector Spaces 119
3.1 Vectors in 2-Space, 3-Space, and n-Space 119
3.2 Norm, Dot Product, and Distance in Rn130
3.3 Orthogonality 143
3.4 The Geometry of Linear Systems 152
3.5 Cross Product 161
CHAPTER 4 General Vector Spaces 171
4.1 Real Vector Spaces 171
4.2 Subspaces 179
4.3 Linear Independence 190
4.4 Coordinates and Basis 200
4.5 Dimension 209
4.6 Change of Basis 217
4.7 Row Space, Column Space, and Null Space 225
4.8 Rank, Nullity, and the Fundamental Matrix Spaces 237
4.9 Matrix Transformations from Rn to Rm 247
4.10 Properties of Matrix Transformations 263
4.11 Application: Geometry of Matrix Operators on R2 273
4.12 Application: Dynamical Systems and Markov Chains 282
CHAPTER 5 Eigenvalues and Eigenvectors 295
5.1 Eigenvalues and Eigenvectors 295
5.2 Diagonalization 305
5.3 Complex Vector Spaces 315
5.4 Application: Differential Equations 327
CHAPTER 6 Inner Product Spaces 335
6.1 Inner Products 335
6.2 Angle and Orthogonality in Inner Product Spaces 345
6.3 Gram-Schmidt Process; QR-Decomposition 352
6.4 Best Approximation; Least Squares 366
6.5 Application: Least Squares Fitting to Data 376
6.6 Application: Function Approximation; Fourier Series 382
CHAPTER 7 Diagonalization and Quadratic Forms 389
7.1 Orthogonal Matrices 389
7.2 Orthogonal Diagonalization 397
7.3 Quadratic Forms 405
7.4 Optimization Using Quadratic Forms 417
7.5 Hermitian, Unitary, and Normal Matrices 424
CHAPTER 8 Linear Transformations 433
8.1 General Linear Transformations 433
8.2 Isomorphism 445
8.3 Compositions and Inverse Transformations 452
8.4 Matrices for General Linear Transformations 458
8.5 Similarity 468
CHAPTER 9 Numerical Methods 477
9.1 LU-Decompositions 477
9.2 The Power Method 487
9.3 Application: Internet Search Engines 496
9.4 Comparison of Procedures for Solving Linear Systems 501
9.5 Singular Value Decomposition 506
9.6 Application: Data Compression Using Singular Value Decomposition 514
APPENDIX A How to Read Theorems 519
APPENDIX B Complex Numbers 521
Answers to Exercises 529
Index 559
