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