Strang: Introduction to Linear Algebra

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Table of Contents

1 Introduction to Vectors 1

1.1 Vectors and Linear Combinations 2

1.2 Lengths and Dot Products 11

1.3 Matrices 22

2 Solving Linear Equations 31

2.1 Vectors and Linear Equations 31

2.2 The Idea of Elimination 46

2.3 Elimination Using Matrices 58

2.4 Rules for Matrix Operations 70

2.5 Inverse Matrices 83

2.6 Elimination = Factorization: A = LU 97

2.7 Transposes and Permutations 109

3 Vector Spaces and Subspaces 123

3.1 Spaces of Vectors 123

3.2 The Nullspace of A: Solving Ax = 0 and Rx = 0 135

3.3 The Complete Solution to Ax = b 150

3.4 Independence, Basis and Dimension 164

3.5 Dimensions of the Four Subspaces 181

4 Orthogonality 194

4.1 Orthogonality of the Four Subspaces 194

4.2 Projections 206

4.3 Least Squares Approximations 219

4.4 Orthonormal Bases and Gram-Schmidt 233

5 Determinants 247

5.1 The Properties of Determinants 247

5.2 Permutations and Cofactors 258

5.3 Cramer’s Rule, Inverses, and Volumes 273

6 Eigenvalues and Eigenvectors 288

6.1 Introduction to Eigenvalues 288

6.2 Diagonalizing a Matrix 304

6.3 Systems of Differential Equations 319

6.4 Symmetric Matrices 338

6.5 Positive Definite Matrices 350

7 The Singular Value Decomposition (SVD) 364

7.1 Image Processing by Linear Algebra 364

7.2 Bases and Matrices in the SVD 371

7.3 Principal Component Analysis (PCA by the SVD) 382

7.4 The Geometry of the SVD 392

8 Linear Transformations 401

8.1 The Idea of a Linear Transformation 401

8.2 The Matrix of a Linear Transformation 411

8.3 The Search for a Good Basis 421

9 Complex Vectors and Matrices 430

9.1 Complex Numbers 431

9.2 Hermitian and Unitary Matrices 438

9.3 The Fast Fourier Transform 445

10 Applications 452

10.1 Graphs and Networks 452

10.2 Matrices in Engineering 462

10.3 Markov Matrices, Population, and Economics 474

10.4 Linear Programming 483

10.5 Fourier Series: Linear Algebra for Functions 490

10.6 Computer Graphics 496

10.7 Linear Algebra for Cryptography 502

11 Numerical Linear Algebra 508

11.1 Gaussian Elimination in Practice 508

11.2 Norms and Condition Numbers 518

11.3 Iterative Methods and Preconditioners 524

12 Linear Algebra in Probability & Statistics 535

12.1 Mean, Variance, and Probability 535

12.2 Covariance Matrices and Joint Probabilities 546

12.3 Multivariate Gaussian and Weighted Least Squares 555

Matrix Factorizations 563

Index 565

Six Great Theorems / Linear Algebra in a Nutshell 574