Linear Algebra

Chapter 1 Preliminaries
1.1 Fields _2
1.2 Matrices and Matrix Operations _8

Chapter 2 Vector Spaces
2.1 Vector Spaces _24
2.2 Vectors in Euclidean Spaces _32
2.3 Subspaces _43
2.4 Bases of Vector Spaces _52

Chapter 3 Systemof Linear Equations
3.1 Gauss-Jordan Elimination Method _74
3.2 Inverse Matrix _103
3.3 Elementary Matrix Multiplications _115
3.4 Row and Column Spaces _125

Chapter 4 Linear Transformations andMatrices
4.1 Linear Transformations _142
4.2 Matrix Representations of Linear Transformations _156
4.3 Compositions of Linear Transformations and Matrices _166
4.4 Change of Basis _172

Chapter 5 Determinants
5.1 Definition of Determinant _182
5.2 Properties of Determinant _191
5.3 Cramer’s Rule _201

Chapter 6 Inner Products
6.1 Inner Products _210
6.2 Gram-Schmidt Theorem _220

Chapter 7 Eigenvalues and Their Applications
7.1 Cayley-Hamilton Theorem _236
7.2 Eigenvalues and Their Applications _244
7.3 Diagonalization of Square matrices _258
7.4 Symmetric Matrices _273
7.5 Quadratic Forms _283

Chapter 8 JordanCanonical Forms
8.1 Jordan Chains _301
8.2 Jordan Canonical Forms _315

References _357

Index _359