Table of Contents

liner-algebra-and-its-applications4thedtion

词汇

elimination
n. 消除（消元）
subtract
v. 减去
substitution
v. 代入
approaches
n. 方法
ratio
n. 比例

Skeleton

Cover

Title Page

Contents

Preface

1 Matrices and Gaussian Elimination

1.1 Introduction

这里 Determinants 的一大坨式子是根据这个公式写的 \[ x = \frac{e \cdot d - b \cdot f}{a \cdot d - b \cdot c} \]

\[ y = \frac{a \cdot f - e \cdot c}{a \cdot d - b \cdot c} \]

1.2 The Geometry of Linear Equations
1.3 An Example of Gaussian Elimination
1.4 Matrix Notation and Matrix Multiplication
1.5 Triangular Factors and Row Exchanges
1.6 Inverses and Transposes
1.7 Special Matrices and Applications
Review Exercises

2 Vector Spaces

2.1 Vector Spaces and Subspaces
2.2 Solving Ax = 0 and Ax = b
2.3 Linear Independence, Basis, and Dimension
2.4 The Four Fundamental Subspaces
2.5 Graphs and Networks
2.6 Linear Transformations
Review Exercises

3 Orthogonality

3.1 Orthogonal Vectors and Subspaces
3.2 Cosines and Projections onto Lines
3.3 Projections and Least Squares
3.4 Orthogonal Bases and Gram-Schmidt
3.5 The Fast Fourier Transform
Review Exercises

4 Determinants

4.1 Introduction
4.2 Properties of the Determinant
4.3 Formulas for the Determinant
4.4 Applications of Determinants
Review Exercises

5 Eigenvalues and Eigenvectors

5.1 Introduction
5.2 Diagonalization of a Matrix
5.3 Difference Equations and Powers Ak
5.4 Differential Equations and eAt
5.5 Complex Matrices
5.6 Similarity Transformations
Review Exercises

6 Positive Definite Matrices

6.1 Minima, Maxima, and Saddle Points
6.2 Tests for Positive Definiteness
6.3 Singular Value Decomposition
6.4 Minimum Principles
6.5 The Finite Element Method

7 Computations with Matrices

7.1 Introduction
7.2 Matrix Norm and Condition Number
7.3 Computation of Eigenvalues
7.4 Iterative Methods for Ax = b

8 Linear Programming and Game Theory

8.1 Linear Inequalities
8.2 The Simplex Method
8.3 The Dual Problem
8.4 Network Models
8.5 Game Theory

A Intersection, Sum, and Product of Spaces

A.1 The Intersection of Two Vector Spaces
A.2 The Sum of Two Vector Spaces
A.3 The Cartesian Product of Two Vector Spaces
A.4 The Tensor Product of Two Vector Spaces
A.5 The Kronecker Product A⊗B of Two Matrices

B The Jordan Form

C Matrix Factorizations

D Glossary: A Dictionary for Linear Algebra

E MATLAB Teaching Codes

F Linear Algebra in a Nutshell

Solutions to Selected Exercises

Problem Set 1.2
Problem Set 1.3
Problem Set 1.4
Problem Set 1.5
Problem Set 1.6
Problem Set 1.7
Problem Set 2.1
Problem Set 2.2
Problem Set 2.3
Problem Set 2.4
Problem Set 2.5
Problem Set 2.6
Problem Set 3.1
Problem Set 3.2
Problem Set 3.3
Problem Set 3.4
Problem Set 3.5
Problem Set 4.2
Problem Set 4.3
Problem Set 4.4
Problem Set 5.1
Problem Set 5.2
Problem Set 5.3
Problem Set 5.4
Problem Set 5.5
Problem Set 5.6
Problem Set 6.1
Problem Set 6.2
Problem Set 6.3
Problem Set 6.4
Problem Set 6.5
Problem Set 7.2
Problem Set 7.3
Problem Set 7.4
Problem Set 8.1
Problem Set 8.2
Problem Set 8.3
Problem Set 8.4
Problem Set 8.5
Problem Set A
Problem Set B

Skeleton

Cover

Title Page

Contents

Preface

1 Matrices and Gaussian Elimination

1.1 Introduction
1.2 The Geometry of Linear Equations
1.3 An Example of Gaussian Elimination
1.4 Matrix Notation and Matrix Multiplication
1.5 Triangular Factors and Row Exchanges
1.6 Inverses and Transposes
1.7 Special Matrices and Applications
Review Exercises

2 Vector Spaces

2.1 Vector Spaces and Subspaces
2.2 Solving Ax = 0 and Ax = b
2.3 Linear Independence, Basis, and Dimension
2.4 The Four Fundamental Subspaces
2.5 Graphs and Networks
2.6 Linear Transformations
Review Exercises

3 Orthogonality

3.1 Orthogonal Vectors and Subspaces
3.2 Cosines and Projections onto Lines
3.3 Projections and Least Squares
3.4 Orthogonal Bases and Gram-Schmidt
3.5 The Fast Fourier Transform
Review Exercises

4 Determinants

4.1 Introduction
4.2 Properties of the Determinant
4.3 Formulas for the Determinant
4.4 Applications of Determinants
Review Exercises

5 Eigenvalues and Eigenvectors

5.1 Introduction
5.2 Diagonalization of a Matrix
5.3 Difference Equations and Powers Ak
5.4 Differential Equations and eAt
5.5 Complex Matrices
5.6 Similarity Transformations
Review Exercises

6 Positive Definite Matrices

6.1 Minima, Maxima, and Saddle Points
6.2 Tests for Positive Definiteness
6.3 Singular Value Decomposition
6.4 Minimum Principles
6.5 The Finite Element Method

7 Computations with Matrices

7.1 Introduction
7.2 Matrix Norm and Condition Number
7.3 Computation of Eigenvalues
7.4 Iterative Methods for Ax = b

8 Linear Programming and Game Theory

8.1 Linear Inequalities
8.2 The Simplex Method
8.3 The Dual Problem
8.4 Network Models
8.5 Game Theory

A Intersection, Sum, and Product of Spaces

A.1 The Intersection of Two Vector Spaces
A.2 The Sum of Two Vector Spaces
A.3 The Cartesian Product of Two Vector Spaces
A.4 The Tensor Product of Two Vector Spaces
A.5 The Kronecker Product A⊗B of Two Matrices

B The Jordan Form

C Matrix Factorizations

D Glossary: A Dictionary for Linear Algebra

E MATLAB Teaching Codes

F Linear Algebra in a Nutshell

Solutions to Selected Exercises

Problem Set 1.2
Problem Set 1.3
Problem Set 1.4
Problem Set 1.5
Problem Set 1.6
Problem Set 1.7
Problem Set 2.1
Problem Set 2.2
Problem Set 2.3
Problem Set 2.4
Problem Set 2.5
Problem Set 2.6
Problem Set 3.1
Problem Set 3.2
Problem Set 3.3
Problem Set 3.4
Problem Set 3.5
Problem Set 4.2
Problem Set 4.3
Problem Set 4.4
Problem Set 5.1
Problem Set 5.2
Problem Set 5.3
Problem Set 5.4
Problem Set 5.5
Problem Set 5.6
Problem Set 6.1
Problem Set 6.2
Problem Set 6.3
Problem Set 6.4
Problem Set 6.5
Problem Set 7.2
Problem Set 7.3
Problem Set 7.4
Problem Set 8.1
Problem Set 8.2
Problem Set 8.3
Problem Set 8.4
Problem Set 8.5
Problem Set A
Problem Set B

Author: verdant

Created: 2026-06-28 Sun 10:50

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