Skip to main content
Linear Algebra
For Undergraduate Students
Yilan(Peter) Tan
Contents
Search Book
close
Search Results:
No results.
Dark Mode
Prev
Up
Next
\(\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
1
Preface
1
System of linear equations
1.1
Motivation: How to Solve a Linear System
1.1.1
Three equation Operations
1.2
Linear Systems and Equivalent Systems
1.3
Gaussian-Jordan Elimination
1.3.1
Gaussian-Jordan Elimination in Linear System
1.3.1
Exercises
1.3.2
A More Compact Representation: Matrices
1.3.2
Exercises
1.4
General Solutions
1.4.1
Finding the General Solution via Row Reduction
1.5
Matrix Form and Vector Form
1.5.1
From Linear System to Matrix Equation
1.5.2
From Vector Equation to Linear System
1.5.3
One Problem, Three Languages
1.6
Computations With SageMath
1.6.1
How to input a regular matrix
1.6.2
How to input a matrix with variables
1.7
Application: Polynomial Curve Fitting
1.7.1
Example 1: Quadratic Through Three Points
1.7.2
Example 2: Polynomial Through Five Points
2
Matrix Operations
2.1
Special Types of Matrices
2.1.1
Zero and Identity Matrices
2.1.2
Diagonal and Scalar Matrices
2.1.3
Triangular Matrices
2.1.4
Symmetric Matrices
2.1.5
Summary and Relationships
2.2
Matrix Addition and Scalar Multiplication
2.2.1
Matrix Addition
2.2.2
Scalar Multiplication
2.2.3
Algebraic Properties
2.2.4
Real-World Applications
2.2.5
Practice Problems
2.3
Matrix Multiplication
2.3.1
Matrix Multiplication
2.3.1
Examples
2.3.2
\(AB\neq BA\)
2.3.3
The
\(n\)
-th Power of a matrix
2.3.3
Exercises
2.4
Inverse Matrix
2.4.1
Theoretical Foundation of Matrix Inversion
2.4.2
Computing Inverse Matrices
2.4.3
Application: Encoding Messages with Invertible Matrices
2.4.3.1
The Encoding Process
2.4.3.2
The Decoding Process
2.4.3.3
Key Observations
2.5
Elementary matrix
2.5.1
Elementary Matrices and Their Properties
2.5.1
Exercises
3
Determinant
3.1
The Formal Definition of Determinant
3.1.1
Determinants of
\(2 \times 2\)
and
\(3 \times 3\)
Matrices
3.1.2
Minors and Cofactors
3.1.3
The Formal Definition: Cofactor Expansion Along the First Row
3.1.4
The Laplace Expansion Theorem
3.2
The Second Way to Compute Determinant
3.2.1
Fast way to get the determinant
3.2.2
Elementary Row Operations and Determinants
3.2.3
Geometric Interpretation of Determinants
3.2.3.1
2D Case: Area of a Parallelogram
3.2.3.2
3D Case: Volume of a Parallelepiped
3.3
Properties of Determinants
3.3.1
Basic Properties
3.3.2
Application: The Adjoint Matrix
3.4
Cramer’s Rule
4
Vector Space
4.1
Vector Space
\(\mathbb{R}^n\)
4.1.1
Span
4.1.2
Linear Combinations and Linear Independence
4.1.3
Basis
4.2
Subspaces of
\(\mathbb{R}^n\)
4.2.1
Definition and Basic Properties
4.2.2
Examples of Subspaces
4.2.3
Important Theorem
4.3
Abstract Vector Spaces
4.3.1
Definition of Vector Space
4.3.2
Examples of Vector Spaces
4.3.3
Basis of Abstract Vector Spaces
4.3.4
Subspaces of Abstract Vector Spaces
4.4
Three Fundamental Subspaces of a Matrix
4.4.1
Definitions and Basic Properties
4.4.2
Looking Ahead: Bases and Rank
4.4.3
Preliminary Exploration
4.4.4
Finding Bases for Row, Column, and Null Spaces
4.5
Application: Coordinate Vectors
4.5.1
Definition of Coordinate Vectors
4.5.2
Properties of Coordinate Vectors
4.5.3
Change of Basis and Transition Matrices
4.5.4
Conclusion
5
Inner Product Space
5.1
Euclidean Space
5.1.1
Dot product in
\(\mathbb{R}^{n}\)
5.1.1
Exercises
5.1.2
Length
5.1.3
Angle
5.1.4
Distance
5.1.5
Cross Product
5.2
Orthogonal Projection
5.2.1
Orthogonal and Orthonormal Sets
5.2.2
Orthogonal Projection
5.3
Gram-Schmidt Process
5.3.1
The Gram-Schmidt Algorithm
5.3.2
Examples of Gram-Schmidt Process
5.4
Inner Product Spaces
5.4.1
Examples of Abstract Vector Spaces
5.4.2
Definition of Inner Product Space
5.4.3
Gram-Schmidt Process in an Inner Product Space
5.5
Least Squares Approximation
5.5.1
Motivation: The Inconsistent System Problem
5.5.2
Geometric Interpretation
5.5.3
The Normal Equations
5.5.4
Examples and Applications
5.5.5
Applications: Mathematical Modeling
5.5.6
Exercises
6
Linear Transformation
6.1
The definition and its matrix of a Linear Transformation
6.1.1
Linear Transformation
6.1.2
Linear Transformation
\(T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}\)
6.1.2
Exercises
6.1.3
Further Reading: The Big Picture
6.2
Understand the linear transformation
6.2.1
Matrix relative to bases
6.2.2
The Fundamental Relation
6.3
The Kernel and range of
\(T\)
6.4
Properties of Linear Transformations
6.4.1
One-to-One and Onto Transformations
6.4.2
Invertible Linear Transformations
6.4.3
Composition of Linear Transformations
6.5
Similar Matrices
6.5.1
Definition and Basic Properties
6.5.2
Optional: Matrix Representations in Different Bases
6.5.3
The Significance of Similarity
7
Eigen-theory
7.1
Eigenvalues and Eigenvectors: Finding Special Directions
7.1.1
Definition of Eigenvalues and Eigenvectors
7.1.2
The Characteristic Polynomial
7.1.3
Eigenspaces
7.1.4
Algebraic and Geometric Multiplicities
7.1.5
Property of Eigenvectors
7.2
Diagonalization: Cracking the Walnut Open
7.2.1
What Does It Mean to Diagonalize?
7.2.2
Why Diagonalization Matters: Computing its Powers
7.3
Orthogonal Diagonalization of Symmetric Matrices
7.3.1
Orthogonal Matrices
7.3.2
Special Properties of Symmetric Matrices
7.4
Quadratic Forms
7.5
Applications I: Differential Equations
7.6
Application II: Computing Matrix Exponentials
7.6.1
Defining the Matrix Exponential
7.7
Application III: Conic Sections, Quadric Surfaces, and Positive Definite Matrices
7.7.1
Conic Sections in
\(\mathbb{R}^2\)
7.7.2
From Conics to Quadric Surfaces
7.7.3
Using Orthogonal Diagonalization to Identify Quadric Surfaces
7.7.4
Positive Definite Matrices
8
Analytic Geometry in Three-Dimensional Space
8.1
9.1 Planes in Space and Their Equations
8.1.1
Point-Normal Form of a Plane
8.1.2
General Form and Geometric Meaning of Coefficients
8.1.3
Intercept Form of a Plane
8.2
9.2 Lines in Space and Their Equations
8.2.1
Parametric Equation of a Line
8.2.2
Symmetric (Point-Direction) Form
8.2.3
General Form: A Line as Intersection of Two Planes
8.3
9.3 Position Relations of Lines and Planes
8.3.1
Plane-Plane Relations
8.3.2
Line-Line Relations
8.3.3
Line-Plane Relations
8.4
9.4 Angles and Distance Formulas in Space
8.4.1
Angle Between a Line and a Plane
8.4.2
Angle Between Two Planes
8.4.3
Distance Formulas in Space
8.5
Chapter Summary
Backmatter
A
PreTeXt 教学环境参考
A.1
内容块环境
A.1.1
例子类环境
A.1.2
定理类环境
A.1.3
定义类环境
A.1.4
注释类环境
A.1.5
汇总类环境
A.2
结构组件
A.2.1
解答组件
A.2.2
证明组件
A.3
交互式元素
A.4
使用建议
A.5
本书使用示例
Colophon
Appendix
A
PreTeXt 教学环境参考
本附录介绍 PreTeXt 中可用的各种教学环境类型。这些环境帮助作者以结构化的方式组织教学内容，并为读者提供清晰的视觉提示。
🔗
A.1
内容块环境
A.2
结构组件
A.3
交互式元素
A.4
使用建议
A.5
本书使用示例
PreTeXt 提供了丰富的环境类型，使作者能够以语义化的方式组织内容。选择合适的环境不仅能改善文档的结构和可读性，还能充分利用 PreTeXt 的交互功能和多种输出格式的特性。
🔗
更多信息请参考
PreTeXt 作者指南
1
pretextbook.org/doc/guide/html/guide-toc.html
。
🔗
🔗
