Introduction to Linear Algebra, Fourth Edition

Table of Contents......Page 4
Preface......Page 6
1: Introduction to Vectors......Page 12
1.1 Vectors and Linear Combinations......Page 13
Review of the key ideas......Page 17
Problem Set 1.1......Page 19
1.2 Lengths and Dot Products......Page 22
1.3 Matrices......Page 33
2.1 Vectors and Linear Equations......Page 42
2.2 The Idea of Elimination......Page 56
2.3 Elimination Using Matrices......Page 68
2.4 Rules for Matrix Operations......Page 79
2.5 Inverse Matrices......Page 93
2.6 Elimination = Factorization: A=LU......Page 107
2.7 Transposes and Permutations......Page 119
3.1 Spaces of Vectors......Page 131
3.2 The Nullspace of A: Solving Ax = 0......Page 143
3.3 The Rank and the Row Reduced Form......Page 155
3.4 The Complete Solution to Ax = b......Page 166
3.5 Independence, Basis and Dimension......Page 179
3.6 Dimensions of the Four Subspaces......Page 195
4.1 Orthogonality of the Four Subspaces......Page 206
4.2 Projections......Page 217
4.3 Least Squares Approximations......Page 229
4.4 Orthogonal Bases and Gram-Schmidt......Page 241
5.1 The Properties of Determinants......Page 255
5.2 Permutations and Cofactors......Page 266
5.3 Cramer\'s Rule, Inverses, and Volumes......Page 280
6.1 Introduction to Eigenvalues......Page 294
6.2 Diagonalizing a Matrix......Page 309
6.3 Applications to Differential Equations......Page 323
6.4 Symmetric Matrices......Page 341
6.5 Positive Definite Matrices......Page 353
6.6 Similar Matrices......Page 366
6.7 Singular Value Decomposition (SVD)......Page 374
7.1 The Idea of a Linear Transformation......Page 386
7.2 The Matrix of a Linear Transformation......Page 395
7.3 Diagonalization and the Pseudoinverse......Page 410
8.1 Matrices in Engineering......Page 420
8.2 Graphs and Networks......Page 431
8.3 Markov Matrices, Population, and Economics......Page 442
8.4 Linear Programming......Page 451
8.5 Fourier Series: Linear Algebra for Functions......Page 458
8.6 Linear Algebra for Statistics and Probability......Page 464
8.7 Computer Graphics......Page 470
9.1 Gaussian Elimination in Practice......Page 476
9.2 Norms and Condition Numbers......Page 486
9.3 Iterative Methods and Preconditioners......Page 492
10.1 Complex Numbers......Page 504
10.2 Hermitian and Unitary Matrices......Page 512
10.3 The Fast Fourier Transform......Page 520
Solutions to Selected Exercises......Page 527
Conceptual Questions for Review......Page 563
Glossary: A Dictionary for Linear Algebra......Page 568
Matrix Factorizations......Page 575
Teaching Codes......Page 577
Index......Page 578
Linear Algebra in a Nutshell......Page 585