# 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.

## Table of Contents

This book is part of a series on Algebra:
Basic Algebra Algebra Intermediate Algebra Vectors Introductory Linear Algebra Linear Algebra Abstract Algebra Intro to Rings and Algebras Associative Composition Algebra
* Cover
* Notation
* Introduction

### Linear Systems

* Solving Linear Systems (Jul 13, 2009)
** Gauss' Method (Jul 13, 2009)
** Describing the Solution Set (Jul 13, 2009)
** General = Particular + Homogeneous (Jul 13, 2009)
** Comparing Set Descriptions (Jul 13, 2009)
** Automation (Jul 13, 2009)
* Linear Geometry of n-Space (Jul 13, 2009)
** Vectors in Space (Jul 13, 2009)
** Length and Angle Measures (Jul 13, 2009)
* Reduced Echelon Form (Jul 13, 2009)
** Gauss-Jordan Reduction (Jul 13, 2009)
** Row Equivalence (Jul 13, 2009)
* Topic: Computer Algebra Systems (Jul 13, 2009)
* Topic: Input-Output Analysis (Jul 13, 2009)
* Input-Output Analysis M File (Mar 24 2008)
* Topic: Accuracy of Computations (Jul 13, 2009)
* Topic: Analyzing Networks (Jul 13, 2009)
* Topic: Speed of Gauss' Method (Mar 24, 2008)

### Vector Spaces  (Apr 17, 2009)

* Definition of Vector Space (Apr 17, 2009)
** Definition and Examples (Jun 18, 2009)
** Subspaces and Spanning sets (Jun 18, 2009)
* Linear Independence (Apr 17, 2009)
** Definition and Examples (Apr 17, 2009)
* Basis and Dimension (Apr 17, 2009)
** Basis (Jun 18, 2009)
** Dimension (Apr 17, 2009)
** Vector Spaces and Linear Systems (Apr 17, 2009)
** Combining Subspaces (Apr 17, 2009)
* Topic: Fields (Apr 17, 2009)
* Topic: Crystals (Apr 17, 2009)
* Topic: Voting Paradoxes (Apr 17, 2009)
* Topic: Dimensional Analysis (Apr 17, 2009)

### Maps Between Spaces

* Isomorphisms (Jun 21, 2009)
** Definition and Examples (July 19, 2009)
** Dimension Characterizes Isomorphism (Jun 21, 2009)
* Homomorphisms (Jun 21, 2009)
** Definition of Homomorphism (Jun 21, 2009)
** Rangespace and Nullspace (Jun 21, 2009)
* Computing Linear Maps (Jun 21, 2009)
** Representing Linear Maps with Matrices (Jun 21, 2009)
** Any Matrix Represents a Linear Map (Jun 21, 2009)
* Matrix Operations (Jun 21, 2009)
** Sums and Scalar Products (Jun 21, 2009)
** Matrix Multiplication (Jun 21, 2009)
** Mechanics of Matrix Multiplication (Jun 21, 2009)
** Inverses (Jun 21, 2009)
* Change of Basis (Jun 21, 2009)
** Changing Representations of Vectors (Jun 21, 2009)
** Changing Map Representations (Jun 21, 2009)
* Projection (Jun 21, 2009)
** Orthogonal Projection Onto a Line (Jun 21, 2009)
** Gram-Schmidt Orthogonalization (Jun 21, 2009)
** Projection Onto a Subspace (Jun 21, 2009)
* Topic: Line of Best Fit (Jun 21, 2009)
* Topic: Geometry of Linear Maps (Jun 21, 2009)
* Topic: Markov Chains (Jun 21, 2009)
* Topic: Orthonormal Matrices (Jun 21, 2009)

### Determinants (Jun 21, 2009)

* Definition (Jun 21, 2009)
** Exploration (Jun 21, 2009)
** Properties of Determinants (Jun 21, 2009)
** The Permutation Expansion (Jun 21, 2009)
** Determinants Exist (Jun 21, 2009)
* Geometry of Determinants (Jun 21, 2009)
** Determinants as Size Functions (Jun 21, 2009)
* Other Formulas for Determinants (Jun 21, 2009)
** Laplace's Expansion (Jun 21, 2009)
* Topic: Cramer's Rule (Jun 21, 2009)
* Topic: Speed of Calculating Determinants (Jun 21, 2009)
* Topic: Projective Geometry (Jun 21, 2009)

### Similarity (Jun 24, 2009)

* Complex Vector Spaces (Jun 24, 2009)
** Factoring and Complex Numbers: A Review (Jun 24, 2009)
** Complex Representations (Jun 24, 2009)
* Similarity
** Definition and Examples (Jun 24, 2009)
** Diagonalizability (Jun 24, 2009)
** Eigenvalues and Eigenvectors (Jun 24, 2009)
* Nilpotence (Jun 24, 2009)
** Self-Composition (Jun 24, 2009)
** Strings (Jun 24, 2009)
* Jordan Form (Jun 24, 2009)
** Polynomials of Maps and Matrices (Jun 24, 2009)
** Jordan Canonical Form (Jun 24, 2009)
* Topic: Geometry of Eigenvalues (Jun 24, 2009)
* Topic: The Method of Powers (Jun 24, 2009)
* Topic: Stable Populations (Jun 24, 2009)
* Topic: Linear Recurrences (Jun 24, 2009)

### Unitary Transformations

* Inner product spaces
* Unitary and Hermitian matrices
* Singular Value Decomposition
* Spectral Theorem

### Appendix

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

### Resources and Licensing

* Licensing And History
* Resources
* Bibliography (see individual pages for references)
* Index (Apr 22, 2010)