線性代數（英文維基教科書）

Linear Algebra 線性代數

An Introduction to Mathematical Discourse 數學話語導論

　　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.
　　這本書是專門為那些以前沒有接觸過數學的學生設計的，因為他們是數學家。也就是說，作為一個目標是從清晰一致的定義開始嚴格證明定理的學科。這本書試圖建立學生從一個背景，數學只是一個工具，提供有用的計算點，學生有一個清晰和精確的數學性質的掌握。引言中對本書的先決條件和目標進行了更詳細的討論。

這本教科書翻譯自英文維基教科書Linear Algebra,相關的翻譯問題可見討論頁。

Table of Contents 目錄

• Cover 封面
• Notation 符號說明
• Introduction 介紹

Linear Systems 線性方程組

1. Solving Linear Systems 求解線性方程組（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