Advanced Algebra (Abstract Part)

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Author: Ren Beishang
Language: English
ISBN/ISSN: 9787560980300
Published on: 2012-01
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Chapter 1Linear Spaces(线性空间)(1)
1.1 Basic Concept(基本概念)(1)
1.1.1 Integers (整数)(1)
1.1.2 Mappings (映射)(3)
1.1.3 Equivalence Relation (等价关系)(6)
1.1.4 Exercises and Supplementary Exercises(习题及补充练习)(7)
1.2 Definition, Examples and Simple Properties of Linear Spaces(线性空间的定义、例子和简单性质)(8)
1.2.1 Definition and Examples of Linear Space (线性空间的定义和例子)(8)
1.2.2 Properties of Iinear Space (线性空间的性质)(10)
1.2.3 Exercises and Supplementary Exercises(习题及补充练习)(11)
1.3 Dimension, Basis and Coordinates (维数、基与坐标)(14)
1.3.1 Linear Combination and Linear Dependence (线性组合及线性相关)(14)
1.3.2 Basis and Dimension of Linear Space(线性空间的基与维数)(15)
1.3.3 Coordinate of a Vector with Respect to the Basis(向量关于基的坐标)(17)
1.3.4 Exercises and Supplementary Exercises(习题及补充练习) (18)
1.4 Basis Change and Coordinate Transformations(基变换与坐标变换)(20)
1.4.1 Basis Change (基变换)(20)
1.4.2 Coordinate Transformations(坐标变换)(22)
1.4.3 The Properties of the Transition Matrix(过渡矩阵的性质)(23)
1.4.4 Exercises and Supplementary Exercises(习题及补充练习)(27)
1.5 Linear Subspaces (线性子空间)(29)
1.5.1 Definition and Examples of Linear Subspace (线性子空间的定义和例子)(29)
1.5.2 Linear Subspaces Generated by a Set of Vectors (由向量组生成的线性子空间)(31)
1.5.3 Intersection Subspace and Sum Subspace (交子空间与和子空间)(33)
1.5.4 Direct Sum of Subspaces (子空间的直和)(36)
1.5.5 Exercises and Supplementary Exercises(习题及补充练习)(39)
1.6 Isomorphism of Linear Spaces (线性空间的同构)(40)
1.6.1 Definition and Simple Properties of Isomorphism of Linear Spaces (线性空间同构的定义和简单性质)(40)
1.6.2 The Application of Isomorphism of Linear Spaces (线性空间同构的应用)(43)
1.6.3 Exercises(习题)(46)
1.7 ※Factor Spaces (商空间)(46)
1.7.1 Properties of Cosets (陪集的性质)(46)
1.7.2 Factor Space (商空间)(48)
Test for Chapter 1 (第1章测试卷)(49)
Biography of A. L. Cauchy(53)
Chapter 2Linear Transformations (线性变换)(54)
2.1 Definition and Operation of Linear Transformation(线性变换的定义和运算)(54)
2.1.1 Definition,Examples and Basic Properties of Linear Transformation (线性变换的定义、范例及基本性质)(54)
2.1.2 Operation of Linear Transformations (线性变换的运算)(56)
2.1.3 The Image and Kernel of a Linear Transformation(线性变换的像与核)(59)
2.1.4 Exercises and Supplementary Exercises(63)
2.2 The Matrix of a Linear Transformation (线性变换的矩阵)(65)
2.2.1 Matrix of a Linear Transformation with Respect to the Basis (线性变换关于基的矩阵)(65)
2.2.2 The Correspondence Relation Between the Linear Transformation and the Matrix (线性变换与矩阵之间的对应关系)(67)
2.2.3 The Relationship between the Coordinates of a Vector and Its Image(向量与它的像的坐标之间的关系)(71)
2.2.4 Exercises and Supplementary Exercises(77)
2.3 Invariant Subspaces (不变子空间)(80)
2.3.1 Definition and Examples of Invariant Subspace (不变子空间的定义和例子)(81)
2.3.2 The Relationship between the Invariant Subspace and Simplified Matrix (不变子空间与化简矩阵的关系)(82)
2.3.3 Exercises and Supplementary Exercises(85)
2.4 Eigenvalues and Eigenvectors (特征值及特征向量)(87)
2.4.1 Concept of Eigenvalues and Eigenvectors of a Linear Transformation (线性变换的特征值和特征向量的概念)(87)
2.4.2 Method for Finding the Eigenvalues and Eigenvectors (特征值和特征向量的求法)(89)
2.4.3 The Eigenvectors of A and Asubspaces(A的特征向量及A子空间)(93)
2.4.4 Exercises and Supplementary Exercises(96)
Test for Chapter 2 (第2章测试卷)(98)
Biography of A.Cayley(102)
Chapter 3Euclidean Spaces(欧几里得空间)(103)
3.1 Concept of Euclidean Spaces (欧几里得空间的概念)(103)
3.1.1 Definition and Examples of Euclidean Spaces (欧几里得空间的定义及实例)(103)
3.1.2 Basic Properties of Euclidean Spaces (欧几里得空间的基本性质)(105)
3.1.3 Exercises and Supplementary Exercises(112)
3.2 Orthonormal Bases (标准正交基)(114)
3.2.1 Orthogonal Set, Orthonormal Set, Orthogonal Basis and Orthonormal basis(正交组,标准正交组,正交基及标准正交基)(114)
3.2.2 Existence of the Orthonormal Basis and Schmidt Orthogonalization Procees(标准正交基的存在性与施密特正交化过程)(120)
3.2.3 The Isomorphism of Euclidean Spaces(欧几里得空间的同构)(123)
3.2.4 Exercises and Supplementary Exercises(124)
3.3 Orthogonal and Symmetric Linear Transformations(正交线性变换及对称线性变换)(126)
3.3.1 Orthogonal Linear Transformations (正交线性变换)(127)
3.3.2 Symmetric Linear Transformations (对称线性变换)(130)
3.3.3 Exercises and Supplementary Exercises(131)
3.4 Orthogonal Complement of Subspaces(子空间的正交补) (134)
3.4.1 Definition and Properties of the Orthogonal Complement of Subspaces(子空间的正交补的定义和性质)(134)
3.4.2 Exercises and Supplementary Exercises(136)
3.5 ※Conjugate Linear Transformations and Unitary Spaces(共轭线性变换及酉空间)(138)
3.5.1 Conjugate Linear Transformations (共轭线性变换)(138)
3.5.2 Unitary Spaces (酉空间)(140)
3.5.3 Exercises and Supplementary Exercises(147)
Test for Chapter 3 (第3章测试卷)(148)
Biography of Euclid(152)
Chapter 4Matrices Similar to Diagonal Matrices(矩阵相似于对角形)(153)
4.1 Diagonalization of Matrices (矩阵的对角化)(153)
4.1.1 Eigenvalues, Eigenvectors and Characteristic Polynomials of a Matrix(矩阵的特征值、特征向量及特征多项式)(153)
4.1.2 Concept of Diagonalization for Matrices (矩阵对角化的概念)(158)
4.1.3 The Relationship between the Diagonalization of A and A(矩阵A与线性变换A的对角化之间的关系)(162)
4.1.4 Exercises and Supplementary Exercises(165)
4.2 Diagonalization of Real Symmetric Matrices and Symmetric Transformations(实对称矩阵及对称变换的对角化)(167)
4.2.1 Basic Properties and Theorems(基本性质和基本定理)(167)
4.2.2 Diagonalization of Real Symmetric Matrices and Symmetric Transformations(实对称矩阵及对称变换的对角化)(169)
4.2.3 Examples (范例)(173)
4.2.4 Exercises and Supplementary Exercises(174)
4.3 CayleyHamilton Theorem and Minimum Polynomial(凯莱哈密尔顿定理及最小多项式)(176)
4.3.1 Cayley  Hamilton Theorem (凯莱哈密尔顿定理)(176)
4.3.2 Minimum Polynomials (最小多项式)(178)
4.3.3 Exercises and Supplementary Exercises(183)
Test for Chapter 4 (第4章测试卷)(185)
Biography of C. Hermite(188)
Chapter 5Jordan Canonical Form ofMatrices(矩阵的若当标准形)(190)
5.1 Invariant Factor, Determinant Division and Condition for Matrices to be Similar(不变因子、行列式因子及矩阵相似的条件)(190)
5.1.1 Necessary and Sufficient Condition for Two Matrices to be Similar(两个矩阵相似的充分必要条件)(190)
5.1.2 Invariant Factor, Determinant Division and Canonical form of λMatrices(不变因子、行列式因子及λ矩阵的标准形)(194)
5.1.3 Exercises and Supplementary Exercises(199)
5.2 Elementary Divisor and Jordan Canonical Forms (初等因子及若当标准形)(201)
5.2.1 Necessary and Sufficient Condition for Two λMatrices to be Equivalent(两个α矩阵等价的充分必要条件)(201)
5.2.2 Basic Properties and Application of Jordan Canonical Forms(若当标准形的基本性质及应用)(206)
5.2.3 ※Rational Canonical Forms of the Matrices (矩阵的有理标准形)(211)
5.2.4 Exercises and Supplementary Exercises(213)
Test for Chapter 5 (第5章测试卷)(216)
Biography of C. Jordan(219)
Chapter 6Quadratic Forms (二次型)(221)
6.1 Standard Forms of General Quadratic Forms(二次型的标准形)(221)
6.1.1 The Matrix Expression of Quadratic Forms and Linear Substitution of Variables (二次型的矩阵表示以及变量的线性代换)(222)
6.1.2 Equivalence of Quadratic Forms and Congruence of Matrices (二次型的等价及矩阵的合同)(224)
6.1.3 Sum of Squares and Standard Forms of Quadratic Forms (二次型的平方和与标准形)(224)
6.1.4 Exercises and Supplementary Exercises(229)
6.2 Properties and Classification of Real Quadratic Forms(实二次型的性质及分类)(231)
6.2.1 Standard Forms of Real Quadratic Forms(实二次型的标准形)(231)
6.2.2 Classification of Real Quadratic Forms (实二次型的分类)(235)
6.2.3 Another Method for Determining of the Positive Definiteness and the Negative Definiteness of a Real Quadratic Form (确定实二次型的正定性和负定性的其他方法)(237)
6.2.4 Exercises and Supplementary Exercises(240)
Test for Chapter 6 (第6章测试卷)(241)
Biography of P.S.Laplace(245)
Chapter 7Bilinear Functions (双线性函数)(247)
7.1 Linear Mappings (线性映射)(247)
7.1.1 Definition, Examples and Basic Properties of Linear Mapping (线性映射的定义、范例和基本性质)(247)
7.1.2 The Restriction and Extension of a Linear Mapping (线性映射的限制及扩张)(252)
7.1.3 The Universal Properties of a Linear Mapping (线性映射的泛性质)(253)
7.1.4 Direct Sum of Linear Spaces and Linear Mappings (线性空间和线性映射的直和)(256)
7.1.5 Exercises and Supplementary Exercises(258)
7.2 Bilinear Functions(双线性函数)(260)
7.2.1 Linear Functions (线性函数)(260)
7.2.2 Bilinear Functions (双线性函数)(260)
7.2.3 Exercises and Supplementary Exercises(264)
7.3 Dual Spaces (对偶空间)(266)
7.3.1 Dual Spaces (对偶空间)(266)
7.3.2 Dual Mappings (对偶映射)(269)
7.3.3 Exercises and Supplementary Exercises(272)
Test for Chapter 7 (第7章测试卷)(274)
Biography of L.Kronecker(278)
Index(中英文名词索引)(279)
Bibliography(283)AOE

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