本書采用學(xué)生易于接受的知識(shí)結(jié)構(gòu)方式和英語表述方式，科學(xué)、系統(tǒng)地介紹了線性代數(shù)的行列式、矩陣、高斯消元法解線性方程組、向量、方程組解的結(jié)構(gòu)、特征值和特征向量、二次型等知識(shí)。強(qiáng)調(diào)通用性和適用性，兼顧先進(jìn)性。本書起點(diǎn)低，難度坡度適中，語言簡(jiǎn)潔明了，不僅適用于課堂教學(xué)使用，同時(shí)也適用于自學(xué)自習(xí)。全書有關(guān)鍵詞索引，習(xí)題按小節(jié)配置，題量適中，題型全面，書后附有答案。
本書讀者對(duì)象為高等院校理工、財(cái)經(jīng)、醫(yī)藥、農(nóng)林等專業(yè)大學(xué)生和教師，特別適合作為中外合作辦學(xué)的國(guó)際教育班的學(xué)生以及準(zhǔn)備出國(guó)留學(xué)深造學(xué)子的參考書。
本書可以作為大學(xué)數(shù)學(xué)線性代數(shù)雙語或英語教學(xué)教師和準(zhǔn)備出國(guó)留學(xué)深造學(xué)子的參考書。特別適合中外合作辦學(xué)的國(guó)際教育班的學(xué)生，能幫助他們較快地適應(yīng)全英文的學(xué)習(xí)內(nèi)容和教學(xué)環(huán)境，完成與國(guó)外大學(xué)學(xué)習(xí)的銜接。本書在定稿之前已在多個(gè)學(xué)校作為校本教材試用，而且得到了師生的好評(píng)。
The authors are pleased to see the text of Linear Algebra in English version for Chinese students at the university level. This book not only shows and explains the useful and beautiful knowledge of mathematics, but also presents the structure and arrangement of linear algebra.
1. The Significance of this Book
“One sows a seed in the spring, thousands of grains autumn to him brings.” All the Chinese students had strict training step by step in the study of mathematics before they become a university student. Intuitive and experimental methods are basic and important study patterns, but the target of mathematical education is to form and improve the deductive ability. So far, Chinese students have distinct and excellent achievement in international comparison of mathematics all over the world. As the improvement in educational exchange internationally, more and more Chinese students choose to study abroad at their university level or higher level. Therefore, mathematical textbook on the basis of Chinese students’ mathematical study in English version is urgent needed and essential. This book provides the strong support for the students who will study Economics, Finance, Management, Social Science and so on in local country or abroad.
2. The Difference between Linear Algebra and Calculus
Calculus is mostly about symmetric and beautiful things.One is differentiation, another is its inverse—integration. Calculus can help us to solve the problems in continuous and analog situation in our life. How about other discrete and digital things? Linear algebra can give us help, and vector and matrix are the second type of language we need to study and understand. Study to read a matrix is the most meaningful and key goal in linear algebra, and it gives wide variety for this mathematical area. There are three examples given here:Triangular Matrix，Symmetric Matrix，Orthogonal Matrix.
3. The Structure of this Book
This book organizes the content basis on the logical relationship among number, matrix and vector. It lists the structure from determinant, to matrix, to solve system of linear equations, to vector, to structure of solutions, to eigenvalue and eigenvector, to quadratic form finally.
Here is the structure of this book:
Chapter 1 starts with determinant. There are three important points about the determinant. The first is the definition, the second is property, and the last is its expansion. The Cramer’s rule is given basis on these three points.
Chapter 2 gives all the varieties of matrix. After the study of concept of matrix, it begins with algebra operations, and shows some special matrices. It is following with how to partition matrix, and how to find the inverse of matrix. After given the elementary operations and elementary matrix, this chapter is ended by rank of matrix.
Chapter 3 shows the relationship between matrix and the system of linear equations. Certainly, it is the most important that using matrix to solve the system of linear equations. Gaussian Elimination Method is the most helpful technique.
Chapter 4 begins studying vector. Definition and operation are two basic study points. Linear dependence and rank of vector are two new knowledge structures.
Chapter 5 is mainly basis on chapter 3 and chapter 4. Here is similar framework for giving the structure of solutions of homogeneous and nonhomogeneous system of linear equations. Both these two parts discuss the corresponding property firstly, and give the details of their structure respectively.
Chapter 6 is mostly in eigenvalue and eigenvector. Besides the definition of them, there are three points of matrix using both two of them which are diagonalization, similar matrix and real symmetric matrix.
Chapter 7 is quadratic form which has three points. The first point is about the definition. The first is the basic, almost, which is the principal and organization order of studying mathematical knowledge. The second is the classification of quadratic form and positive definite matrix. The last is criterion of congruent matrix.
4. Help with this Book
“Not knowing that flower close to the water earlier blow, I wonder if it’s last winter’s unmelted snow.” This textbook is emerged with the strong support from Applied Mathematical Department of BNUZ firstly, and the cooperation of senior professor and junior lecture in warm，selfless and enthusiastic environment. Certainly, it has very close relationship with the developing and open international education in BNUZ. Thank you all.
毛綱源，武漢理工大學(xué)資深教授，畢業(yè)于武漢大學(xué)，留校任教，后調(diào)入武漢工業(yè)大學(xué)（現(xiàn)合并為武漢理工大學(xué)）擔(dān)任數(shù)學(xué)物理系系主任，在高校從事數(shù)學(xué)教學(xué)與科研工作40余年，除了出版多部專著（早在1998年，世界科技出版公司W(wǎng)orld Scientific Publishing Company就出版過他主編的線性代數(shù)Linear Algebra的英文教材）和發(fā)表數(shù)十篇專業(yè)論文外，還發(fā)表10余篇考研數(shù)學(xué)論文。
主講微積分、線性代數(shù)、概率論與數(shù)理統(tǒng)計(jì)等課程。理論功底深厚，教學(xué)經(jīng)驗(yàn)豐富，思維獨(dú)特。曾多次受邀在各地主講考研數(shù)學(xué)，得到學(xué)員的廣泛認(rèn)可和一致好評(píng)：“知識(shí)淵博，講解深入淺出，易于接受”“解題方法靈活，技巧獨(dú)特，輔導(dǎo)針對(duì)性極強(qiáng)”“對(duì)考研數(shù)學(xué)的出題形式、考試重點(diǎn)難點(diǎn)了如指掌，上他的輔導(dǎo)班受益匪淺”。
馬迎秋，北京師范大學(xué)珠海分校副教授，畢業(yè)于渤海大學(xué)，愛爾蘭都柏林大學(xué)數(shù)學(xué)碩士。主講微積分、線性代數(shù)、數(shù)學(xué)教學(xué)論、數(shù)學(xué)教學(xué)設(shè)計(jì)、數(shù)學(xué)史與數(shù)學(xué)文化等課程。在國(guó)內(nèi)外權(quán)wei期刊發(fā)表中英文論文10余篇。
梁敏，北京師范大學(xué)珠海分校副教授，畢業(yè)于天津大學(xué)，美國(guó)托萊多大學(xué)數(shù)學(xué)碩士，美國(guó)羅格斯大學(xué)統(tǒng)計(jì)學(xué)碩士。主講微積分、線性代數(shù)、概率論與數(shù)理統(tǒng)計(jì)、商務(wù)統(tǒng)計(jì)、運(yùn)籌學(xué)等課程。在國(guó)內(nèi)外權(quán)wei期刊發(fā)表中英文論文10余篇。
Chapter 1 Determinant（1）
1.1 Definition of Determinant（1）
1.1.1 Determinant arising from the solution of linear system（1）
1.1.2 The definition of determinant of order n（5）
1.1.3 Determine the sign of each term in a determinant （8）
Exercise 1.1（10）
1.2 Basic Properties of Determinant and Its Applications（12）
1.2.1 Basic properties of determinant（12）
1.2.2 Applications of basic properties of determinant（15）
Exercise 1.2（19）
1.3 Expansion of Determinant （21）
1.3.1 Expanding a determinant using one row (column)（21）
1.3.2 Expanding a determinant along k rows (columns)（27）
Exercise 1.3（29）
1.4 Cramer’s Rule（30）
Exercise 1.4（36）
Chapter 2 Matrix（38）
2.1 Matrix Operations（38）
2.1.1 The concept of matrices（38）
2.1.2 Matrix Operations（41）
Exercise 2.1（58）
2.2 Some Special Matrices（60）
Exercise 2.2（64）
2.3 Partitioned Matrices（66）
Exercise 2.3（72）
2.4 The Inverse of Matrix（73）
2.4.1 Finding the inverse of an n×n matrix（73）
2.4.2 Application to economics（81）
2.4.3 Properties of inverse matrix （83）
2.4.4 The adjoint matrix A (or adjA) of A（86）
2.4.5 The inverse of block matrix（89）
Exercise 2.4（91）
2.5 Elementary Operations and Elementary Matrices（94）
2.5.1 Definitions and properties （94）
2.5.2 Application of elementary operations and elementary matrices（100）
Exercise2.5（102）
2.6 Rank of Matrix（103）
2.6.1 Concept of rank of a matrix（104）
2.6.2 Find the rank of matrix（107）
Exercise 2.6（109）
Chapter 3 Solving Linear System by Gaussian Elimination Method（110）
3.1 Solving Nonhomogeneous Linear System by Gaussian Elimination Method（110）
3.2 Solving Homogeneous Linear Systems by Gaussian Elimination Method（128）
Exercise 3（131）
Chapter 4 Vectors（134）
4.1 Vectors and its Linear Operations（134）
4.1.1 Vectors（134）
4.1.2 Linear operations of vectors（136）
4.1.3 A linear combination of vectors （137）
Exercise 4.1（143）
4.2 Linear Dependence of a Set of Vectors （143）
Exercise 4.2（155）
4.3 Rank of a Set of Vectors（156）
4.3.1 A maximal independent subset of a set of vectors（156）
4.3.2 Rank of a set of vectors（159）
Exercise 4.3（163）
Chapter 5 Structure of Solutions of a System（165）
5.1 Structure of Solutions of a System of Homogeneous Linear Equations （165）
5.1.1 Properties of solutions of a system of homogeneous linear equations（165）
5.1.2 A system of fundamental solutions （166）
5.1.3 General solution of homogeneous system（171）
5.1.4 Solutions of system of equations with given solutions of the system（173）
Exercise 5.1（176）
5.2 Structure of Solutions of a System of Nonhomogeneous Linear Equations（178）
5.2.1 Properties of solutions（178）
5.2.2 General solution of nonhomogeneous equations （179）
5.2.3 The simple and convenient method of finding the system of fundamental solutions and particular solution（183）
Exercise 5.2（189）
Chapter 6 Eigenvalues and Eigenvectors of Matrices（191）
6.1 Find the Eigenvalue and Eigenvector of Matrix（191）
Exercise 6.1（197）
6.2 The Proof of Problems Related with Eigenvalues and Eigenvectors（198）
Exercise 6.2（199）
6.3 Diagonalization（200）
6.3.1 Criterion of diagonalization（200）
6.3.2 Application of diagonalization（209）
Exercise 6.3（210）
6.4 The Properties of Similar Matrices（211）
Exercise 6.4（216）
6.5 Real Symmetric Matrices（218）
6.5.1 Scalar product of two vectors and its basis properties（218）
6.5.2 Orthogonal vector set（220）
6.5.3 Orthogonal matrix and its properties（223）
6.5.4 Properties of real symmetric matrix（225）
Exercise 6.5（229）
Chapter 7 Quadratic Forms （231）
7.1 Quadratic Forms and Their Standard Forms（231）
Exercise 7.1（236）
7.2 Classification of Quadratic Forms and Positive Definite Quadratic(Positive Definite Matrix)（237）
7.2.1 Classification of Quadratic Form（237）
7.2.2 Criterion of a positive definite matrix（239）
Exercise 7.2（241）
7.3 Criterion of Congruent Matrices（242）
Exercise 7.3（245）
Answers to Exercises（246）
Appendix Index（266）
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