本书叙述深入浅出，以矩阵为主线，突出矩阵的运算和化简，突出用矩阵方法研究线性方程组、二次型和实际问题模型。本书对于抽象的理论和方法，总是从具体问题入手，再将其推广到一般情形，而略去了许多繁杂的理论推导，并力求将数学与应用相结合。
本书的主要内容包括线性方程组、矩阵代数、行列式、向量空间、矩阵的特征值与特征向量和二次型等。
本书是一本介绍性的线性代数教材，内容简洁，层次清晰，适合高等学校理工科专业线性代数课程双语教学使用。

The matrix is the mainline of the book. With the help of the matrix operation and the matrix simplification, we study the linear equations, the quadratic forms and the real world applications. For the purpose of the insights into the abstract theory and the methods of the linear algebra, we start to discuss the conceptions and the methods with the specific problems, then we directly extend them to the general situation without the complicated theoretical derivation. Furthermore, we try to combine the mathematical methods with the real applications in this book.
The main contents of the book are linear equations, matrix algebra, determinants, vector spaces, eigenvalues and eigenvectors，and quadratic forms, etc.

Chapter 1 Linear Equations in Linear Algebra001
1.1Systems of Linear Equations001
1.2Row Reduction and Echelon Forms008
1.3Solutions of Linear Systems012
1.4Vector Equations014
Exercises017

Chapter 2 Matrix Algebra019
2.1Matrix Operations019
2.2The Inverse of a Matrix024
2.3Partitoned Matrices028
2.4Matrix Factorizations031
2.5Subspace of Rn032
2.6Dimension and Rank035
Exercises037

Chapter 3 Determinants040
3.1Introduction to Determinants040
3.2Properties of Determinants043
3.3Cofactor Expansion048
3.4The Inverse of a Matrix050
3.5Cramer’s Rule053
Exercises054

Chapter 4 Vector Spaces058
4.1Definition of Vector Spaces058
4.2Subspaces and Span062
4.3Linearly Independent Sets068
4.4Bases and Dimension071
4.5Inner Product，Length，Angle074
4.6Orthonormal Basis and the Gram-Schmidt Procedure078
Exercises084

Chapter 5 Eigenvalues and Eigenvectors088
5.1Definition of Eigenvalues and Eigenvectors088
5.2Properties of Eigenvalues and Eigenvectors092
5.3Similarity and Diagonalization096
5.4Diagonalization of Symmetric Matrices100
Exercises105

Chapter 6 Solution Sets of Linear Systems107
6.1Homogeneous Linear Systems107
6.2Solutions of Nonhomogeneous Systems108
6.3Applications of Linear Systems110
Exercises113

Chapter 7 Symmetric Matrices and Quadratic Forms117
7.1Diagonalization of Symmetric Matrices117
7.2Quadratic Forms119
7.3Quadratic Problems122
7.4The Singular Value Decomposition126
7.5Applications to Statistics129
Exercises132

References134