主要內容包括：向量代數，線性方程組，矩陣代數，行列式及特征值與特征向量及實對稱矩陣與二次型等內容；每章開始給出與本章內容相關的歷史發展進程，針對相應知識點給出幾何及工程實際應用案例，其中工程實際應用案例主要以不同應用領域的具體問題為驅動，利用相關基本知識進行建模與分析，提供應用線性代數知識解決實際問題的思想，并對重點問題給出具體python算例；習題部分設置一定數量的實際應用問題，可以擴展和加深線性代數知識的理解與應用。
1.黑龍江省自然科學基金，杰出青年基金，JQ2020A001，i-量子群的幾何實現與朗蘭茲對偶，2020-07至2023-07，50萬元，在研，主持。
2. 國家自然科學基金，面上項目，11671108，運用幾何方法研究量子群的典范基和Kazhdan-Lusztig理論，2017-01至2020-12，48萬元，已結題，主持。
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
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