Contents :

1. Linear Equations 1 --
1.2 Gaussian Elimination and Matrices 3 --
1.3 Gauss-Jordan Method 15 --
1.4 Two-Point Boundary Value Problems 18 --
1.5 Making Gaussian Elimination Work 21 --
1.6 Ill-Conditioned Systems 33 --
2. Rectangular Systems and Echelon Forms 41 --
2.1 Row Echelon Form and Rank 41 --
2.2 Reduced Row Echelon Form 47 --
2.3 Consistency of Linear Systems 53 --
2.4 Homogeneous Systems 57 --
2.5 Nonhomogeneous Systems 64 --
2.6 Electrical Circuits 73 --
3. Matrix Algebra 79 --
3.1 From Ancient China to Arthur Cayley 79 --
3.2 Addition and Transposition 81 --
3.3 Linearity 89 --
3.4 Why Do It This Way 93 --
3.5 Matrix Multiplication 95 --
3.6 Properties of Matrix Multiplication 105 --
3.7 Matrix Inversion 115 --
3.8 Inverses of Sums and Sensitivity 124 --
3.9 Elementary Matrices and Equivalence 131 --
3.10 The LU Factorization 141 --
4. Vector Spaces 159 --
4.1 Spaces and Subspaces 159 --
4.2 Four Fundamental Subspaces 169 --
4.3 Linear Independence 181 --
4.4 Basis and Dimension 194 --
4.5 More about Rank 210 --
4.6 Classical Least Squares 223 --
4.7 Linear Transformations 238 --
4.8 Change of Basis and Similarity 251 --
4.9 Invariant Subspaces 259 --
5. Norms, Inner Products, and Orthogonality 269 --
5.1 Vector Norms 269 --
5.2 Matrix Norms 279 --
5.3 Inner-Product Spaces 286 --
5.4 Orthogonal Vectors 294 --
5.5 Gram-Schmidt Procedure 307 --
5.6 Unitary and Orthogonal Matrices 320 --
5.7 Orthogonal Reduction 341 --
5.8 Discrete Fourier Transform 356 --
5.9 Complementary Subspaces 383 --
5.10 Range-Nullspace Decomposition 394 --
5.11 Orthogonal Decomposition 403 --
5.12 Singular Value Decomposition 411 --
5.13 Orthogonal Projection 429 --
5.14 Why Least Squares? 446 --
5.15 Angles between Subspaces 450 --
6. Determinants 459 --
6.1 Determinants 459 --
6.2 Additional Properties of Determinants 475 --
7. Eigenvalues and Eigenvectors 489 --
7.1 Elementary Properties of Eigensystems 489 --
7.2 Diagonalization by Similarity Transformations 505 --
7.3 Functions of Diagonalizable Matrices 525 --
7.4 Systems of Differential Equations 541 --
7.5 Normal Matrices 547 --
7.6 Positive Definite Matrices 558 --
7.7 Nilpotent Matrices and Jordan Structure 574 --
7.8 Jordan Form 587 --
7.9 Functions of Nondiagonalizable Matrices 599 --
7.10 Difference Equations, Limits, and Summability 616 --
7.11 Minimum Polynomials and Krylov Methods 642 --
8. Perron-Frobenius Theory 661 --
8.2 Positive Matrices 663 --
8.3 Nonnegative Matrices 670 --
8.4 Stochastic Matrices and Markov Chains 687.