This book introduces the concepts and techniques of linear algebra, including systems of linear equations, determinants, matrices, vectors, eigenvalues, eigenvectors and similar matrices. In addition, some topics in applied linear algebra are also provided. Each chapter is followed by plentiful exercises along with reference answers. This book aims not only to help students understand basic definitions and theorems, but to introduce to them how to solve linear algebra problems with MATLAB. The goal is to give the freshman students a solid understanding of the basic ideas, as well as an appreciation of how they are used in real world applications and how Linear Algebra is a rigorous mathematical subject itself.
This book is suitable for bilingual teaching and can serve as English teaching materials for international students studying in China. It can also be used as a textbook for Linear Algebra course for English majors, as well as a reading reference book for students majoring in applied mathematics, and information and computing science.
Table of Contents
Chapter 1 Systems of Linear Equations and Elementary Operations on Matrices
1．1 Systems of Linear Equations
1．1．1 Definition of Systems of Linear Equations
1．1．2 Equivalent Systems and Gaussian Elimination
Exercises 1．1
1．2 Elementary Operations on Matrices
1．2．1 Elementary Row Operations
1．2．2 Row Echelon Forms
1．2．3 Standard Form of a Matrix
1．2．4 Analyzing Networks
Exercises 1．2
Chapter 2 Determinants
2．1 The Determinant of a Square Matrix
2．1．1 Determinants of Order 2 and Order 3
2．1．2 Permutations and Number of Inversions
2．1．3 Determinants of Order n
Exercises 2．1
2．2 Properties of Determinants
Exercises 2．2
2．3 Laplace Expansion of a Determinant
Exercises 2．3
2．4 Applications of Determinants
2．4．1 Cramer's Rule
2．4．2 Geometric Applications of Determinants
Exercises 2．4
Chapter 3 Matrices
3．1 Matrix Arithmetic
3．1．1 Matrices
3．1．2 Matrix Addition
3．1．3 Scalar Multiplication
3．1．4 Matrix Multiplication
3．1．5 Matrix Powers
Exercises 3．1
3．2 Special Matrices
3．2．1 The Identity Matrix
3．2．2 Diagonal Matrices
3．2．3 Triangular Matrices
3．2．4 The Transpose of a Matrix
3．2．5 Symmetric and Skew-symmetric Matrices
3．2．6 The Determinant of the Product AB
3．2．7 The Adjoint of a Square Matrix
Exercises 3．2
3．3 The Inverse of a Matrix
3．3．1 Definition of the Inverse of a Matrix
3．3．2 Properties of Invertible Matrices
Exercises 3．3
3．4 Partitioned Matrices
3．4．1 Definition of Partitioned Matrices
3．4．2 Operations on Partitioned Matrices
Exercises 3．4
3．5 Elementary Matrices
3．5．1 Definition of Elementary Matrices
3．5．2 The Fundamental Theorem of Invertible Matrices
3．5．3 The Gauss-Jordan Method -for Finding the Inverses
Exercises 3．5
3．6 The Rank of a Matrix
3．6．1 Definition of the Rank of a Matrix
3．6．2 Properties of the Ranks of Matrices
Exercises 3．6
3．7 Applications of Matrices
3．7．1 Markov Chains
3．7．2 Linear Economic Models
3．7．3 Networks and Graphs
Exercises 3．7
……
Chapter 4 Structures of Solutions of Linear Systems
Chapter 5 Eigenvalues and Eigenvectors， Sinular Matrices
Chapter 6 Solving Linear Algebra Problems with MATLAB
