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Chap 1 Linear equations and matrix algebra 1

1-1 System of linear equations 2

1-2 Elimination 7

1-3 Elimination using matrix multiplications 12

1-4 Matrix algebra 17

1-5 Inverse matrices 22

1-6 Finding the inverse 26

1-7 LU factorization 29

1-8 Transposes 33

1-9 Block matrices 37

Chap 2 Vector spaces 41

2-1 Vector spaces and subspaces 42

2-2 Column space and nullspace 47

2-3 Reduced row echelon form 52

2-4 General solutions to Ax=b 56

2-5 Independence,basis and dimension 61

2-6 Four fundamental subspaces 66

2-7 Existence and uniqueness of inverses 70

Course Review 1（Chap1～2） 75

Chap 3 Linear transformations 81

3-1 Introduction to Linear transformations 82

3-2 Language of transformations 88

3-3 Coordinate systems and general vector spaces 93

3-4 Matrices of linear transformations 99

3-5 Change of basis 105

Chap 4 Orthogonality 110

4-1 Orthogonal vectors and orthogonal subspaces 111

4-2 Projections 116

4-3 Least squares approximations 121

4-4 Orthogonal matrices 125

4-5 Gram-Schmidt Process 130

Chap 5 Determinants 135

5-1 Properties of determinants 136

5-2 Formulas for deteminants 141

5-3 Applications of determinants 145

Course Review 2（Chap3～5） 150

Chap 6 Eigenanalysis 155

6-1 Eigenvalues and eigenvectors 156

6-2 Properties of eigenvalues and eigenvectors 161

6-3 Diagonalization 167

6-4 Difference equations 171

6-5 Differential equations 177

6-6 Complex eigenvalues 181

6-7 Similar matrices 186

Chap 7 Quadratic forms 191

7-1 Symmetric matrices 192

7-2 Introduction to quadratic forms 197

7-3 Positive definite matrices 202

7-4 Singular values 207

7-5 Singular value decomposition 211

Course Review 3（Chap6～7） 217

Review of Linear Algebra 221