Linear Algebra

本书是一本适合我国数学类本科生特点且顺应二十世纪代数学发展方向的英文高等代数教材，主要有以下几个特色：1.内容编排自然合理，逻辑性强，教材结构更加符合代数学的历史发展进程。以如何求解线性方程组为出发点，进而考虑解的结构，自然引申出向量、矩阵、行列式、线性空间等概念并展开讨论。2.强调线性影射的作用。在教材中加强了代数结构及其影射这一现代观点，以研究线性空间这一代数结构及其之间的联系为主线。另外，用矩阵刻划向量空间的线性影射，由向量空间线性影射的运算诱导出矩阵的运算，从而强化了用具体来刻划抽象的数学思

本书用英语写成，包含多项式和线性代数的基本内容，主要包括：整数和多项式，线性方程组，线性映射，矩阵和行列式，线性空间和线性映射，线性变换等。本书适合数学系本科生作为高等代数教材使用，也可作为双语教学和线性代数的参考教材。

1 Integers and Polynomials
1.1 Integers
1.2 N'umber Fields
1.3 Polynomial
1.4 Polynomial Functions and Roots
1.5 Polvnomials over Rational Number Field
1.6 Polynomials of Several Variables
1.7 Symmetric Polynomials
1.8 Exercises
2 Systems of Linear Equations
2.1 Systems of Linear Equations and Elimination
2.2 Vectors
2.3 Matrices
2.4 Structure of Solutions of A System of Linear Equations
2.5 Exercises
3 Linear Maps, Matrices and Determinants
3. 1 Linear Maps of Vector Spaces and Matrices
3.2 Operations of Linear Maps and Matrices
3.3 Partitioned Matrices
3, 4 Elementary Matrices and Invertible Matrices
3.5 Determinants
3.5.1 Permutation and Determinant
3.5.2 Properties of Determinant
3.5.3 Expansion of Determinant
3.5.4 Applications of Determinant
3.6 Exercises
4 Linear Spaces and Linear Maps
4. 1 Linear Spaces
4.2 Dimension, Basis, Coordinates
4.3 Basis Change and Coordinate Transformations
4.4 Linear Maps and Isomorphism
4.5 Matrices of Linear Maps
4.6 Subspaces and Direct Sum
4.7 Space Decomposition and Partitioned Matrices
4.8 Exercises
5 Linear Transformations
5.1 Linear Transformations
5.2 Similarity of Matrices
5.3 ),-Matrices
5.4 Eigenvalues, Eigenvectors and Characteristic Polynomials
5.5 Invariant Subspaces
5.6 Equivalence of λ-matrices
5.7 Invariant Factors and Elementary Divisors
5.8 Condition for Similarity of Matrices
5.9 Jordan Canonical Forms of Matrices
5.10 Rat iona! Canonical Forms of Matrices
5.11 Exercises
6 Euclidean Spaces
6.1 Inner Product and Basic Properties
6.2 Orthogonal Bases and Schmidt Orthonormalization
6.3 Subspaces and Orthogonal Complements
6.4 Isometry and Orthogonal Transformations
6.5 Symmetric Matrices and Symmetric Transformations
6.6 The Method of Least Squares--System of Linear Equations
Revisited
6.7 A Brief Introduction to Unitary Spaces
6.8 Exercises
7 Linear Forms, Bilinear Forms and Quadratic Forms
7.1 Linear Forms and the Dual Space
7.2 Bilinear Forms
7.3 Symmetric Bilinear Forms
7.4 Quadratic Forms
7.5 Quadratic Forms over Real and Complex Number Fields
7.6 Positive Definite Quadratic Forms over Real Number Field
7.7 Exercises
Bibliography
Index