Catelog-1

1 Vectors and Matrices

1.1 Vectors and Linear Compositions . . . . . . . . . . . . . . . . . 2

1.2 Lengths and Angles from Dot Products . . . . . . . . . . . . . . 9

1.3 Matrices and Their Column Spaces . . . . . . . . . . . . . . . . 18

1.4 Matrix Multiplication $AB$ and $CR$ . . . . . . . . . . . . . . . . . 27

2 Solving Linear Equations $Ax = b$

2.1 Elimination and Back Substitution . . . . . . . . . . . . . . . . 39

2.2 Elimination Matrices and Inverse Matrices . . . . . . . . . . . . 40

2.3 Matrix Computations and $A = LU$ . . . . . . . . . . . . . . . . . 49

2.4 Permutations and Transposes . . . . . . . . . . . . . . . . . . . 57

2.5 Derivatives and Finite Difference Matrices . . . . . . . . . . . . 64

3 The Four Fundamental Subspaces

3.1 Vector Spaces and Subspaces . . . . . . . . . . . . . . . . . . . 74

3.2 Computing the Nullspace by Elimination: $A = CR$ . . . . . . . . . 84

3.3 The Complete Solution to $Ax = b$ . . . . . . . . . . . . . . . . . 85

3.4 Independence, Basis, and Dimension . . . . . . . . . . . . . . . 93

3.5 Dimensions of the Four Subspaces . . . . . . . . . . . . . . . . . 104

4 Orthogonality

4.1 Orthogonality of Vectors and Subspaces . . . . . . . . . . . . . 115

4.2 Projections onto Lines and Subspaces . . . . . . . . . . . . . . . 129

4.3 Least Squares Approximations . . . . . . . . . . . . . . . . . . 143

4.4 Orthonormal Bases and Gram-Schmidt . . . . . . . . . . . . . . . 144

4.5 The Pseudoinverse of a Matrix . . . . . . . . . . . . . . . . . . 151

5 Determinants

5.1 3 by 3 Determinants and Cofactors . . . . . . . . . . . . . . . . 163

5.2 Computing and Using Determinants . . . . . . . . . . . . . . . . 176

5.3 Areas and Volumes by Determinants . . . . . . . . . . . . . . . . 190

6 Eigenvalues and Eigenvectors

6.1 Introduction to Eigenvalues: $Ax = λx$ . . . . . . . . . . . . . . . 198

6.2 Diagonalizing a Matrix . . . . . . . . . . . . . . . . . . . . . . 199

6.3 Symmetric Positive Definite Matrices . . . . . . . . . . . . . . . 205

6.4 Complex Numbers and Vectors and Matrices . . . . . . . . . . . . 211

6.5 Solving Linear Differential Equations . . . . . . . . . . . . . . . 232