本書敘述深入淺出，以矩陣為主線，突出矩陣的運算和化簡，突出用矩陣方法研究線性方程組、二次型和實際問題模型。本書對于抽象的理論和方法，總是從具體問題入手，再將其推廣到一般情形，而略去了許多繁雜的理論推導，并力求將數學與應用相結合。
本書的主要內容包括線性方程組、矩陣代數、行列式、向量空間、矩陣的特征值與特征向量和二次型等。
本書是一本介紹性的線性代數教材，內容簡潔，層次清晰，適合高等學校理工科專業線性代數課程雙語教學使用。
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
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