內容大鋼
This book introduces the concepts and techniques of linear algebra, including systems oflinear equations, determinants, matrices, vectors, eigenvalues, eigenvectors and similar ma-trices. In addition, some topics in applied linear algebra are also provided. Each chapter isfollowed by plentiful exercises along with reference answers. This book aims not only to helpstudents understand basic definitions and theorems, but to introduce to them how to solvelinear algebra problems with MATLAB. The goal is to give the freshman students a solidunderstanding of the basic ideas, as well as an appreciation of how they are used in real worldapplications and how Linear Algebra is a rigorous mathematical subject itself.
This book is suitable for bilingual teaching and can serve as English teaching materialsfor international students studying in China. It can also be used as a textbook for Linear Al-gebra course for English majors, as well as a reading reference book for students majoring inapplied mathematics, and information and computing science.
目錄
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
4.1 Vectors and Operations on Vectors
Exercises 4.1
4.2 Existence and Uniqueness of Solutions
Exercises 4.2
4.3 Linear Relation among Vectors
4.3.1 Linear Combinations of Vectors
4.3.2 Linear Dependence and Linear Independence
Exercises 4.3
4.4 Rank of a Vector Set
4.4.1 Equivalence Relation among Vector Sets
4.4.2 Rank of a Vector Set
4.4.3 Relation between the Rank of a Matrix and the Rank of a Vector Set
Exercises 4.4
4.5 Structure of the General Solution of a Linear System
4.5.1 Homogeneous Linear Systems
4.5.2 Nonhomogeneous Linear Systems
Exercises 4.5
4.6 Vector Spaces
4.6.1 Vector Spaces Axioms
4.6.2 Subspaces
4.6.3 Spanning Sets
4.6.4 Bases and Dimensions
4.6.5 Coordinates and Transition Matrices
Exercises 4.6
4.7 Applications of Vectors
Chapter 5 Eigenvalues and Eigenvectors, Similar Matrices
5.1 Eigenvalues and Eigenvectors
5.1.1 Definition of Eigenvalues and Eigenvectors
5.1.2 Properties of the Eigenvalues and Eigenvectors
Exercises 5.1
5.2 Similarity and Diagonalization
5.2.1 Similar Matrices
5.2.2 Diagonalization
Exercises 5.2
5.3 Diagonalization of Symmetric Matrices
5.3.1 Inner (Scalar or Dot) Product of Vectors
5.3.2 Gram-Schmidt Process
5.3.3 Orthogonal Matrices and Orthogonal Transformation
5.3.4 Orthogonal Diagonalization of Symmetric Matrices
Exercises 5.3
5.4 Quadratic Forms
5.4.1 Definition of Quadratic Forms
5.4.2 Diagonalization of Quadratic Forms
5.4.3 Definiteness of Quadratic Forms
Exercises 5.4
5.5 Applications of Eigenvalues
5.5.1 Systems of Linear Differential Equations
5.5.2 Optimization. the Minimum and Maximum of Functions
Exercises 5.5
Chapter 6 Solving Linear Algebra Problems with MATLAB
6.1 Matrix and Vector Operations
6.1.1 Basic Elementary Functions
6.1.2 Addition and Subtraction
6.1.3 Multiplication, Transpose and Powers
6.1.4 Eigenvalues and Eigenvectors
6.1.5 Some MATLAB Details
Exercises 6.1
6.2 Solving Systems of Linear Equations
6.2.1 Solving Linear Systems in Different Ways
6.2.2 Draw Linear Equations in Two or Three Dimensions
Exercises 6.2
6.3 Programming MATLAB
6.3.1 MATLAB Functions
6.3.2 Looping
6.3.3 Logic
Exercises 6.3
Answers to Exercises
References
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