The matrix is the mainline of the book. With the help of the matrix operation and the matrix simplification, we study the linear equations, the quadratic forms and the real world applications. For the purpose of the insights into the abstract theory and the methods of the linear algebra, we start to discuss the conceptions and the methods with the specific problems, then we directly extend them to the general situation without the complicated theoretical derivation. Furthermore, we try to combine the mathematical methods with the real applications in this book.
The main contents of the book are linear equations, matrix algebra, determinants, vector spaces, eigenvalues and eigenvectors，and quadratic forms, etc.
目錄：
Chapter 1 Linear Equations in Linear Algebra001
1.1Systems of Linear Equations001
1.2Row Reduction and Echelon Forms008
1.3Solutions of Linear Systems012
1.4Vector Equations014
Exercises017
Chapter 2 Matrix Algebra019
2.1Matrix Operations019
2.2The Inverse of a Matrix024
2.3Partitoned Matrices028
2.4Matrix Factorizations031
2.5Subspace of Rn032
2.6Dimension and Rank035
Exercises037
Chapter 3 Determinants040
3.1Introduction to Determinants040
3.2Properties of Determinants043
3.3Cofactor Expansion048
3.4The Inverse of a Matrix050
3.5Cramers Rule053
Exercises054
Chapter 4 Vector Spaces058
4.1Definition of Vector Spaces058
4.2Subspaces and Span062
4.3Linearly Independent Sets068
4.4Bases and Dimension071
4.5Inner Product，Length，Angle074
4.6Orthonormal Basis and the Gram-Schmidt Procedure078
Exercises084
Chapter 5 Eigenvalues and Eigenvectors088
5.1Definition of Eigenvalues and Eigenvectors088
5.2Properties of Eigenvalues and Eigenvectors092
5.3Similarity and Diagonalization096
5.4Diagonalization of Symmetric Matrices100
Exercises105
Chapter 6 Solution Sets of Linear Systems107
6.1Homogeneous Linear Systems107
6.2Solutions of Nonhomogeneous Systems108
6.3Applications of Linear Systems110
Exercises113
Chapter 7 Symmetric Matrices and Quadratic Forms117
7.1Diagonalization of Symmetric Matrices117
7.2Quadratic Forms119
7.3Quadratic Problems122
7.4The Singular Value Decomposition126
7.5Applications to Statistics129
Exercises132
References134
內容試閱：
The main goal of the text is to help students master the basic concepts and skills they will use later in their study and careers.The text provides a modern elementary introduction to linear algebra and a broad selection of interesting applications.The material is accessible to students with the maturity that should come from successful completion of two semesters of college-level calculus.
We have attempted to give this book the following distinctive features.
（1）Many fundamental ideas of linear algebra are introduced within the first lectures，then gradually examined from different points of view.A major achievement of the text is that the level of difficulty is fairly even.
（2）Good notation is crucial，and the text reflects the way scientists and engineers actually use linear algebra in practice.The definitions and proofs focus on the columns of a matrix rather than on the matrix entries.This modern approach simplifies many arguments，and it ties vector space ideas into the study of linear systems.
（3）A broad selection of applications illustrates the power of linear algebra to explain fundamental principles and simplify computing in engineering，physics，economics，and statistics.
In this volume，Chapter 1，Chapter 2，Chapter 6 and Chapter 7 are written by Professor Guoqing Liu，Chapter 3 is written by Associated Professor Jian Zhao，and Chapter 4 and Chapter 5 are written by Dr.Wei Shi.All the chapters are checked and revised by Professor Guoqing Liu.
We hope this book can bring readers some help in the studying and teaching of bilingual mathematics.Due to the limit of our ability，it is impossible to avoid some unclear explanations.We would appreciate any constructive criticisms and corrections from readers.
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