线性代数（英文维基教科书）

Linear Algebra 线性代数

An Introduction to Mathematical Discourse 数学话语导论

　　The book was designed specifically for students who had not previously been exposed to mathematics as mathematicians view it. That is, as a subject whose goal is to rigorously prove theorems starting from clear consistent definitions. This book attempts to build students up from a background where mathematics is simply a tool that provides useful calculations to the point where the students have a grasp of the clear and precise nature of mathematics. A more detailed discussion of the prerequisites and goal of this book is given in the introduction.
　　这本书是专门为那些以前没有接触过数学的学生设计的，因为他们是数学家。也就是说，作为一个目标是从清晰一致的定义开始严格证明定理的学科。这本书试图建立学生从一个背景，数学只是一个工具，提供有用的计算点，学生有一个清晰和精确的数学性质的掌握。引言中对本书的先决条件和目标进行了更详细的讨论。

这本教科书翻译自英文维基教科书Linear Algebra,相关的翻译问题可见讨论页。

Table of Contents 目录

• Cover 封面
• Notation 符号说明
• Introduction 介绍

Linear Systems 线性方程组

1. Solving Linear Systems 求解线性方程组（2020年12月9日）
   1. Gauss' Method 高斯消元法
   2. Describing the Solution Set 解集的表示
   3. General = Particular + Homogeneous
   4. Comparing Set Descriptions
   5. Automation
2. Linear Geometry of n-Space
   1. Vectors in Space
   2. Length and Angle Measures
3. Reduced Echelon Form
   1. Gauss-Jordan Reduction
   2. Row Equivalence
4. Topic: Computer Algebra Systems
5. Topic: Input-Output Analysis
6. Input-Output Analysis M File
7. Topic: Accuracy of Computations
8. Topic: Analyzing Networks
9. Topic: Speed of Gauss' Method

Vector Spaces

1. Definition of Vector Space
   1. Definition and Examples
   2. Subspaces and Spanning sets
2. Linear Independence
   1. Definition and Examples
3. Basis and Dimension
   1. Basis
   2. Dimension
   3. Vector Spaces and Linear Systems
   4. Combining Subspaces
4. Topic: Fields
5. Topic: Crystals
6. Topic: Voting Paradoxes
7. Topic: Dimensional Analysis

Maps Between Spaces

1. Isomorphisms
   1. Definition and Examples
   2. Dimension Characterizes Isomorphism
2. Homomorphisms
   1. Definition of Homomorphism
   2. Rangespace and Nullspace
3. Computing Linear Maps
   1. Representing Linear Maps with Matrices
   2. Any Matrix Represents a Linear Map
4. Matrix Operations
   1. Sums and Scalar Products
   2. Matrix Multiplication
   3. Mechanics of Matrix Multiplication
   4. Inverses
5. Change of Basis
   1. Changing Representations of Vectors
   2. Changing Map Representations
6. Projection
   1. Orthogonal Projection Onto a Line
   2. Gram-Schmidt Orthogonalization
   3. Projection Onto a Subspace
7. Topic: Line of Best Fit
8. Topic: Geometry of Linear Maps
9. Topic: Markov Chains
10. Topic: Orthonormal Matrices

Determinants

1. Definition
   1. Exploration
   2. Properties of Determinants
   3. The Permutation Expansion
   4. Determinants Exist
2. Geometry of Determinants
   1. Determinants as Size Functions
3. Other Formulas for Determinants
   1. Laplace's Expansion
4. Topic: Cramer's Rule
5. Topic: Speed of Calculating Determinants
6. Topic: Projective Geometry

Similarity

1. Complex Vector Spaces
   1. Factoring and Complex Numbers: A Review
   2. Complex Representations
2. Similarity
   1. Definition and Examples
   2. Diagonalizability
   3. Eigenvalues and Eigenvectors
3. Nilpotence
   1. Self-Composition
   2. Strings
4. Jordan Form
   1. Polynomials of Maps and Matrices
   2. Jordan Canonical Form
5. Topic: Geometry of Eigenvalues
6. Topic: The Method of Powers
7. Topic: Stable Populations
8. Topic: Linear Recurrences

Unitary Transformations

1. Inner product spaces
2. Unitary and Hermitian matrices
3. Singular Value Decomposition
4. Spectral Theorem

Appendix

• Propositions
• Quantifiers
• Techniques of Proof
• Sets, Functions, Relations

Resources and Licensing

• Licensing And History
• Resources
• Bibliography (see individual pages for references)
• Index