線性代數引論

《線性代數引論》內容覆蓋了我國現行理工科大學線性代數課程的全部內容，與我國現行的線性代數教學大綱和教材體系比較接近。其中包括矩陣與線性方程組、二維和三維空間、向量空間R、特徵值問題、向量空間和線性交換、行列式、特徵值及其應用等。《線性代數引論》的編寫採用模塊式結構，便於廣大教師根據教學需要對內容進行取捨。《線性代數引論》通過例子介紹了非常流行的教學軟件Matlab在線性代數中的應用，並且每章結尾都附有專門用Matlab做的練習題。《線性代數引論》可供理工科、經濟管理各專業學生作為教科書或參考書，也可供科技人員和自學者參考。

1 MATRICES AND SYSTEMS OF LINEAR EQUATIONS

1.1 Introduction to Matrices and Systems of Linear Equations

1.2 Echelon Form and Gauss-Jordan Elimination

1.3 Consistent Systems of Linear Equations

1.4 Applications (Optional)

1.5 Matrix Operations

1.6 Algebraic Properties'of Matrix Operations

1.7 Linear Independence and Nonsingular Matrices

1.8 Data Fitting, Numerical Integration, and Numerical Differentiation (Optional)

1.9 Matrix Inverses and Their Properties

2 VECTORS IN 2-SPACE AND 3-SPACE

2.1 Vectors in the Plane

2.2 Vectors in Space

2.3 The Dot Product and the Cross Product

2.4 Lines and Planes in Space

3 THE VECTOR SPACE Rn

3.1 Introduction

3.2 Vector Space Properties of Rn

3.3 Examples of Subspaces

3.4 Bases for Subspaces

3.5 Dimension

3.6 Orthogonal Bases for Subspaces

3.7 Linear Transformations from Rn to Rm

3.8 Least-Squares Solutions to Inconsistent Systems, with Applications to Data Fitting

3.9 Theory and Practice of Least Squares

4 THE EIGENVALUE PROBLEM

4.1 The Eigenvalue Problem for (2 x 2) Matrices

4.2 Determinants and the Eigenvalue Problem

4.3 Elementary Operations and Determinants (Optional)

4.4 Eigenvalues and the Characteristic Polynomial

4.5 Eigenvectors and Eigenspaces

4.6 Complex Eigenvalues and Eigenvectors

4.7 Similarity Transformations and Diagonalization

4.8 Difference Equations; Markov Chains; Systems of Differential Equations (Optional)

5 VECTOR SPACES AND LINEAR TRANSFORMATIONS

5.1 Introduction

5.2 Vector Spaces

5.3 Subspaces

5.4 Linear Independence, Bases, and Coordinates

5.5 Dimension

5.6 Inner-Product Spaces, Orthogonal Bases, and Projections (Optional)

5.7 Linear Transformations

5.8 Operations with Linear Transformations

5.9 Matrix Representations for Linear Transformations

5.10 Change of Basis and Diagonalization

6.DETERMINANTS

6.1 Introduction

6.2 Cofactor Expansions of Determinants

6.3 Elementary Operations and Determinants

6.4 Cramer's Rule

6.5 Applications of Determinants: Inverses and Wronksians

7.EIGENVALUES AND APPLICATIONS

7.1 Quadratic Forms

7.2 Systems of Differential Equations

7.3 Transformation to Hessenberg Form

7.4 Eigenvalues of Hessenberg Matrices

7.5 Householder Transformations

7.6 The QR Factorization and Least-Squares Solutions

7.7 Matrix Polynomials and the Cayley-Hamilton Theorem

7.8 Generalized Eigenvectors and Solutions of Systems of Differential Equations

APPENDIX: AN INTRODUCTION TO MATLAB

ANSWERS TO SELECTED ODD-NUMBERED EXERCISES

INDEX