Introduction to 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 _33
2.3 Subspaces _45
2.4 Bases of Vector Spaces _54

Chapter 3 System of Linear Equations
3.1 Gauss-Jordan Elimination Method _78
3.2 Inverse Matrix _110
3.3 Elementary Matrix Multiplications _123
3.4 Row and Column Spaces _134

Chapter 4 Linear Transformations and Matrices
4.1 Linear Transformations _152
4.2 Matrix Representations of Linear Transformations _171
4.3 Compositions of Linear Transformations and Their Matrices _184
4.4 Change of Basis _191

Chapter 5 Determinants
5.1 Definition of Determinant _204
5.2 Properties of Determinant _215
5.3 Cramer’s Rule _226

Chapter 6 Inner Products
6.1 Inner Products _236
6.2 Gram-Schmidt Theorem _249

Chapter 7 Eigenvalues and Their Applications
7.1 Cayley-Hamilton Theorem _268
7.2 Eigenvalues and Their Applications _278
7.3 Diagonalization of Square matrices _293
7.4 Diagonalization of Symmetric Matrices _309
7.5 Quadratic Forms _320

Chapter 8 Jordan Canonical Forms
8.1 Jordan Chains _340
8.2 Jordan Canonical Forms _357

Answers to Exercises _407
References _547
Index _549