Introduction to Linear Algebra Fifth Edition

Table of Contents......Page 4
Chapter 1......Page 12
1.1 Vectors and Linear Combinations......Page 13
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 57
2.3 Elimination Using Matrices......Page 69
2.4 Rules for Matrix Operations......Page 81
2.5 Inverse Matrices......Page 94
2.6 Elimination = Factorization: A = LU......Page 108
2.7 Transposes and Permutations......Page 120
3.1 Spaces of Vectors......Page 134
3.2 The Nullspace of A: Solving Ax= 0 and Rx=0......Page 146
3.3 The Complete Solution to Ax = b......Page 161
3.4 Independence, Basis and Dimension......Page 175
3.5 Dimensions of the Four Subspaces......Page 192
4.1 Orthogonality of the Four Subspaces......Page 205
4.2 Projections......Page 217
4.3 Least Squares Approximations......Page 230
4.4 Orthonormal Bases and Gram-Schmidt......Page 244
5.1 The Properties of Determinants......Page 258
5.2 Permutations and Cofactors......Page 269
5.3 Cramer's Rule, Inverses, and Volumes......Page 284
6.1 Introduction to Eigenvalues......Page 299
6.2 Diagonalizing a Matrix......Page 315
6.3 Systems of Differential Equations......Page 330
6.4 Symmetric Matrices......Page 349
6.5 Positive Definite Matrices......Page 361
7.1 Image Processing by Linear Algebra......Page 375
7.2 Bases and Matrices in the SVD......Page 382
7.3 Principal Component Analysis (PCA by the SVD)......Page 393
7.4 The Geometry of the SVD......Page 403
8.1 The Idea of a Linear Transformation......Page 412
8.2 The Matrix of a Linear Transformation......Page 422
8.3 The Search for a Good Basis......Page 432
Chapter 9......Page 441
9.1 Complex Numbers......Page 442
9.2 Hermitian and Unitary Matrices......Page 449
9.3 The Fast Fourier Transform......Page 456
10.1 Graphs and Networks......Page 463
10.2 Matrices in Engineering......Page 473
10.3 Markov Matrices, Population, and Economics......Page 485
10.4 Linear Programming......Page 494
10.5 Fourier Series: Linear Algebra for Function......Page 501
10.6 Computer Graphics......Page 507
10.7 Linear Algebra for Cryptography......Page 513
11.1 Gaussian Elimination in Practice......Page 519
11.2 Norms and Condition Numbers......Page 529
11.3 Iterative Methods and Preconditioners......Page 535
12.1 Mean, Variance, and Probability......Page 546
12.2 Covariance Matrices and Joint Probabilities......Page 557
12.3 Multivariate Gaussian and Weighted Least Squares......Page 566
MATRIX FACTORIZATIONS......Page 574
C......Page 576
D......Page 577
F......Page 578
J......Page 579
N......Page 580
R......Page 581
s......Page 582
z......Page 583