Contents
Chapter 1 Vector Spaces 1
1.1 Introduction 1
1.2 The geometry and algebra of vectors 1
1.3 Operations of vectors and their applications 12
1.4 Lines and planes in 3-dimensional space 28
1.5 Review exercises 35
Chapter 2 Systems of Linear Equations 38
2.1 Introduction 38
2.2 Solutions of linear systems: elimination method 40
2.3 Structure of solutions of linear systems and linear independence 51
2.4 Subspaces of and linear transformation 63
2.5 Applications 69
2.6 Review exercises 80
Chapter 3 Matrix Algebra 85
3.1 Introduction 85
3.2 Definitions and basic operations of matrices 86
3.3 Matrix multiplication 91
3.4 The inverse of a matrix 103
3.5 Elementary matrices 111
3.6 Review exercises 116
Chapter 4 Determinants 120
4.1 Introduction 120
4.2 The definition and properties of determinants 121
4.3 Geometric interpretations of determinants 130
4.4 Applications of determinants 133
4.5 Review exercises 141
Chapter 5 Eigenvalues and Eigenvectors 145
5.1 Introduction 145
5.2 Definitions of eigenvalues and eigenvectors 146
5.3 Properties of eigenvalues and eigenvectors 155
5.4 Eigenvalues and eigenvectors of symmetric matrices 160
5.5 Similarity and diagonalization 169
5.6 Quadratic forms 177
5.7 Applications 185
5.8 Review exercises 188
Answers to Exercises 192
Chapter 1 192
Chapter 2 197
Chapter 3 205
Chapter 4 213
Chapter 5 217
References 229
Index of Vocabulary 230
Index of Notation 233