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 Cramers 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 nn 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 QuadraticPositive 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）
內容試閱：
The authors are pleased to see the text of Linear Algebra in English version for Chinese students at theuniversity level. This booknot 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 BookOne 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 CalculusCalculus is mostly about symmetric and beautiful things.One is differentiation, another is its inverseintegration. 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 Cramers 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 its last winters 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.
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