Advanced Algebra(Abstract Part)

    
高等代数是数学专业的一门重要的基础课程。它以矩阵、向量、线性空间和线性变换作为主要的研究对象，对培养学生的抽象思维能力、逻辑推理能力，以及数学专业的若干后续课程的学习都起着非常重要的作用。
任北上编著的《Advanced Algebra(Abstract
Part)》在每一章我们都选择了大批具有典型意义的例题，帮助学生举一反三，触类旁通，其中有一些就是本课程的重要结论。通过例题的学习，学生不仅可以更容易地理解抽象的数学概念和内容，疏通各知识链条环环相扣的彼此联系，而且更便于加深对课堂内容的吸纳和消化，从中掌握本课程的数学思想和数学方法。

Chapter 1Linear Spaces(线性空间)(1)
 1.1 Basic Concept（基本概念）(1)
  1.1.1 Intege (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 Dimeion, 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 Dimeion 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 Traformatio（基变换与坐标变换）(20)
  1.4.1 Basis Change (基变换)(20)
  1.4.2 Coordinate Traformatio（坐标变换）(22)
  1.4.3 The Properties of the Traition 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 Vecto
(由向量组生成的线性子空间)(31)
  1.5.3 Inteection 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 Traformatio (线性变换)(54)
 2.1 Definition and Operation of Linear
Traformation（线性变换的定义和运算）(54)
  2.1.1 Definition，Examples and Basic Properties of Linear
Traformation (线性变换的定义、范例及基本性质)(54)
  2.1.2 Operation of Linear Traformatio （线性变换的运算）(56)
  2.1.3 The Image and Kernel of a Linear Traformation（线性变换的像与核）(59)
  2.1.4 Exercises and Supplementary Exercises(63)
 2.2 The Matrix of a Linear Traformation (线性变换的矩阵)(65)
  2.2.1 Matrix of a Linear Traformation 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 Traformation
and the Matrix （线性变换与矩阵之间的对应关系）(67)
  2.2.3 The Relatiohip 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 Relatiohip 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 Eigenvecto (特征值及特征向量)(87)
  2.4.1 Concept of Eigenvalues and Eigenvecto of a Linear
Traformation (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 Eigenvecto
(特征值和特征向量的求法)(89)
  2.4.3 The Eigenvecto 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
Traformatio(正交线性变换及对称线性变换)(126)
  3.3.1 Orthogonal Linear Traformatio (正交线性变换)(127)
  3.3.2 Symmetric Linear Traformatio (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 Traformatio and Unitary
Spaces（共轭线性变换及酉空间）(138)
  3.5.1 Conjugate Linear Traformatio (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, Eigenvecto 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 Relatiohip 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
Traformatio（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
Traformatio（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 （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 Functio (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 Exteion of a Linear Mapping
(线性映射的限制及扩张)(252)
  7.1.3 The Univeal 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 Functio(双线性函数)(260)
  7.2.1 Linear Functio (线性函数)(260)
  7.2.2 Bilinear Functio (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

高等代数是数学专业的一门重要的基础课程。它以矩阵、向量、线性空间和线性变换作为主要的研究对象，对培养学生的抽象思维能力、逻辑推理能力，以及数学专业的若干后续课程的学习都起着非常重要的作用。自2001年以来，尤其是*教高［2001］4号文件和46号文件下发后，为了适应当前我国高校培养创新人才的需要，在*积极稳妥的引领和鼓励下，国内几乎所有高校已在相关专业开展双语教学课程，并趋于理性和成熟。但总体来看，双语教学在国内还只是处在实验和探索的起步阶段，有很多问题亟待研究和解决，如双语教学与母语教学以及本国传统文化的关系、双语教学与外语教学的关系、双语教学的目标和标准、双语教学的课程选择、双语教学的师资建设和教材建设、双语教学的模式选择等。近年来，我们陆续开展了“西部高等院校理科专业课程群双语教学及‘双语’师资队伍培养和建设的研究与实践”、“高师院校数学专业课程双语教学的研究与实验”等课题的研究。配合项目的研究，我们在高等代数课程教学实践环节中，也曾挑选和试用了多部英文原版教材。海外教材确实在立体化配套、多种教学资源的整合以及为课程提供整体教学解决方案等方面有不少可供借鉴之处。同时学生在原汁原味的专业英语教学活动中，能更近距离地了解所学的数学对象的时代背景、先进的教学理念；领略国外学者提出和分析问题的思想以及解决问题的思路、方式和应用技巧；在实践专业英语学习的同时，检验自身通过外语学习和掌握新知识的能力。但一个不容忽视的问题是，外版教材与我国现行的教学内容、教学体系、教学模式和习惯存在着巨大的差异。我们自始至终面临着教材的国际化与本民族的文化教育传统相融合的问题。具体而言，由于课时所限而不得不对外文原版教材不断进行改编或调整，试图在教学内容和教学方式上更符合该课程教学大纲的要求。这本高等代数教材就是我们自治区级教学团队经过多年的教学实践，针对上述项目潜心研讨及自治区级精品课程“代数学”建设过程的研究成果之一。双语教学实践表明，地方普通高校新生的专业英语的接受和消化能力较弱。因此，高等代数这门学年所开设的课程可能从下册再开展双语教学活动更适宜些。本着这样的原则，本教材只涵盖了高等代数课程抽象部分的内容。从近四年的连续使用情况来看，学生普遍反映教材章节安排自然合理，逻辑清晰，每一章都选择了一批有一定层次的典型例题进行分析讲解，例题的内容涵盖了相应章节中高等代数常用的数学思想和方法。本书的编者们吸取了多年的教学实践经验、教改研究成果和国外原版优秀教材的长处，本教材有以下特点。(1)
每章前面都设置了知识脉络图解，以框图的方式概括了本章的知识结构，提纲挈领，一目了然。(2)
结合教材的内容分别介绍了有关的历史回顾和有关数学家的生平，将数学文化与数学历史渗透在教材中，以提高学生的学习兴趣，拓展学生的知识视野和培养学生的数学素养。(3)
在每一章我们都选择了大批具有典型意义的例题，帮助学生举一反三，触类旁通，其中有一些就是本课程的重要结论。通过例题的学习，学生不仅可以更容易地理解抽象的数学概念和内容，疏通各知识链条环环相扣、相互关联的彼此联系，而且更便于加深对课堂内容的吸纳和消化、从中掌握本课程的数学思想和数学方法。(4)
考虑到高等代数也是数学专业硕士研究生入学考试的一门必考科目以及全国大学生数学竞赛的内容之一，本教材中几乎每节都配备了约20道习题。前10道一般为基本训练题，主要是为了加深初学者对高等代数中诸多抽象概念的理解，故此选题基本覆盖了该章节的主要内容。后面约10题则作为有志报考硕士研究生和参与全国大学生数学竞赛的学生的学习补充和提高训练。每章后还配置了一份自测试卷，方便任课教师、学生对自身的教、学作阶段性的小结和梳理。(5)
本书是为学习高等代数课程的数学专业学生编写的，然而事实上高等代数的大部分内容都涵盖了线性代数课程的内容，所以本书也适合大学理工科与经济管理学科等相关专业学生在学习线性代数课程时作为参考书。为了满足自学能力较强和希望对代数学加深了解的本科生、理科研究生的学习愿望，本书中还在部分章节中增设附加的选修内容并加上了“※”号，其他初学者可不必为之大伤脑筋。本书的编写始终得到广西师范学院及教务处领导的支持和鼓励，还得到下列各基金项目的资助：国家自然科学基金项目（10961007）、广西自然科学基金项目（2011GXNSFA018144，2010GXNSFB013048）、广西教育厅科研项目（200911MS145）、新世纪广西高等教育教学改革工程项目（2012JGA162）。在编写过程中，我们也参阅了部分中外代数教材，在此特向编者致谢。A类立项本教材是我们自治区级教学团队共同策划、分工协作的成果。具体分工如下：任北上、刘立明编写了第1、2、3、7章；苏华东、黄倩霞、杨立英编写了第4、5章；黄倩霞、任北上编写了第6章。后由任北上、李碧荣统稿，刘立明、李碧荣、冯家佳审校。
Advanced Algebra is an important basic course for students majoring
in Mathematics, with matrix, vector, linear space and linear
transformation as its main research objects. It plays a significant
role in cultivating students’ abilities in abstract thinking and
logical reasoning, and lays a solid foundation for math majors’
learning of some following courses. 　　Since 2001, especially after
the issuing of the 4th document and 46th document by the Department
of Higher Education, Ministry of Education, in order to meet the
nation’s needs for cultivating innovating talents, almost all the
colleges in China, led and encouraged actively and steadily by the
Ministry of Education, have launched bilingual courses in relevant
majors which are becoming increasingly rational and mature. But
generally speaking, bilingual teaching in China is still in its
empirical and exploratory stage with many problems waiting to be
studied and solved, such as the relationship between bilingual
teaching and native language teaching as well as between bilingual
teaching and native culture, the relationship between bilingual
teaching and foreign language teaching, the goals and standards for
bilingual teaching, the course selection for bilingual teaching,
the construction of teaching faculty and textbooks for bilingual
teaching, the selection of the teaching modes and so on. In recent
years, we have conducted research projects such as “The Bilingual
Teaching of Science Courses in Western Colleges of China and the
Research and Practice of Cultivating Bilingual Teaching Faculty”
and “The Research and Experiment of Bilingual Teaching of
Mathematical Curriculum in Colleges”. Besides, we have selected and
used some original English textbooks in the teaching practice of
advanced algebra. Admittedly, these foreign textbooks do have some
advantages in comprehensive design, integration of various kinds of
teaching resources and in providing solutions for the overall
teaching conduct. Othermore, exposed to an authentic English
teaching environment, students can better grasp the era background
of the mathematical objects they are learning and the advanced
teaching concepts, appreciate the mathematical thinking of foreign
scholars to present and analyze problems as well as their
strategies, methods and application skills of solving problems.
They can also check whether they have mastered the new knowledge
and can put what they have learned into practice through