1 Vectors and Matrices
1.1 Vectors and Linear Combinations
1.2 Lengths and Angles from Dot Products
1.3 Matrices and Their Column Spaces
1.4 Matrix Multiplication AB and CR
2 Solving Linear Equations Ax = b
2.1 Elimination and Back Substitution
2.2 Elimination Matrices and Inverse Matrices
2.3 Matrix Computations and A = LU
2.4 Permutations and Transposes
3 The Four Fundamental Subspaces
3.1 Vector Spaces and Subspaces
3.2 Computing the Nullspace by Elimination:A=CR
3.3 The Complete Solution to Ax = b
3.4 Independence, Basis, and Dimension
3.5 Dimensions of the Four Subspaces
4 Orthogonality
4.1 Orthogonality of Vectors and Subspaces
4.2 Projections onto Lines and Subspaces
4.3 Least Squares Approximations
4.4 Orthonormal Bases and Gram-Schmidt
4.5 The Pseudoinverse of a Matrix
5 Determinants
5.1 3 by 3 Determinants and Cofactors
5.2 Computing and Using Determinants
5.3 Areas and Volumes by Determinants
6 Eigenvalues and Eigenvectors
6.1 Introduction to Eigenvalues : Ax = λx
6.2 Diagonalizing a Matrix
6.3 Symmetric Positive Definite Matrices
6.4 Complex Numbers and Vectors and Matrices
6.5 Solving Linear Differential Equations
7 The Singular Value Decomposition (SVD)
7.1 Singular Values and Singular Vectors
7.2 Image Processing by Linear Algebra
7.3 Principal Component Analysis (PCA by the SVD)
8 Linear Transformations
8.1 The Idea of a Linear Transformation
8.2 The Matrix of a Linear Transformation
8.3 The Search for a Good Basis
9 Linear Algebra in Optimization
9.1 Minimizing a Multivariable Function
9.2 Backpropagation and Stochastic Gradient Descent
9.3 Constraints, Lagrange Multipliers, Minimum Norms
9.4 Linear Programming, Game Theory, and Duality
10 Learning from Data
10.1 Piecewise Linear Learning Functions
10.2 Creating and Experimenting
10.3 Mean, Variance, and Covariance
Appendix 1 The Ranks of AB and A + B
Appendix 2 Matrix Factorizations
Appendix 3 Counting Parameters in the Basic Factorizations
Appendix 4 Codes and Algorithms for Numerical Linear Algebra
Appendix 5 The Jordan Form of a Square Matrix
Appendix 6 Tensors
Appendix 7 The Condition Number of a Matrix Problem
Appendix 8 Markov Matrices and Perron-Frobenius
Appendix 9 Elimination and Factorization
Appendix 10 Computer Graphics
Index of Equations
Index of Notations
Index
