主要内容包括：向量代数，线性方程组，矩阵代数，行列式及特征值与特征向量及实对称矩阵与二次型等内容；每章开始给出与本章内容相关的历史发展进程，针对相应知识点给出几何及工程实际应用案例，其中工程实际应用案例主要以不同应用领域的具体问题为驱动，利用相关基本知识进行建模与分析，提供应用线性代数知识解决实际问题的思想，并对重点问题给出具体python算例；习题部分设置一定数量的实际应用问题，可以扩展和加深线性代数知识的理解与应用。

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