Linear Algebra

This book discusses proof-based linear algebra. The book was designed specifically for students who have 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.

Because of the proof-based nature of this book, readers are recommended to be familiar with mathematical proof before reading this book (although this is not a prerequisite, strictly speaking), so that their reading experiences can be smoother. To gain familiarity with mathematical proof and also some basic mathematical concepts, readers may read the wikibook Mathematical Proof. For a milder introduction to linear algebra that is not too proof-based, see the wikibook Introductory Linear Algebra.

• Cover
• Notation
• Introduction

1. Solving Linear Systems
   1. Gauss' Method
   2. Describing the Solution Set
   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

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

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

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

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

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

The following is a brief overview of some basic concepts in mathematics. For more details, reader can read the wikibook Mathematical Proof.

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

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