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

 * Cover
 * Notation
 * Introduction

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

Vector Spaces
 Definition of Vector Space Linear Independence Basis and Dimension Topic: Fields <li>Topic: Crystals <li>Topic: Voting Paradoxes <li>Topic: Dimensional Analysis </ol>
 * 1) Definition and Examples
 * 2) Subspaces and Spanning sets
 * 1) Definition and Examples
 * 1) Basis
 * 2) Dimension
 * 3) Vector Spaces and Linear Systems
 * 4) Combining Subspaces

Maps Between Spaces
<ol type="I"> <li>Isomorphisms <li>Homomorphisms <li>Computing Linear Maps <li>Matrix Operations <li>Change of Basis <li>Projection <li>Topic: Line of Best Fit <li>Topic: Geometry of Linear Maps <li>Topic: Markov Chains <li>Topic: Orthonormal Matrices </ol>
 * 1) Definition and Examples
 * 2) Dimension Characterizes Isomorphism
 * 1) Definition of Homomorphism
 * 2) Rangespace and Nullspace
 * 1) Representing Linear Maps with Matrices
 * 2) Any Matrix Represents a Linear Map
 * 1) Sums and Scalar Products
 * 2) Matrix Multiplication
 * 3) Mechanics of Matrix Multiplication
 * 4) Inverses
 * 1) Changing Representations of Vectors
 * 2) Changing Map Representations
 * 1) Orthogonal Projection Onto a Line
 * 2) Gram-Schmidt Orthogonalization
 * 3) Projection Onto a Subspace

Determinants
<ol type="I"> <li>Definition <li>Geometry of Determinants <li>Other Formulas for Determinants <li>Topic: Cramer's Rule <li>Topic: Speed of Calculating Determinants <li>Topic: Projective Geometry </ol>
 * 1) Exploration
 * 2) Properties of Determinants
 * 3) The Permutation Expansion
 * 4) Determinants Exist
 * 1) Determinants as Size Functions
 * 1) Laplace's Expansion

Similarity
<ol type="I"> <li>Complex Vector Spaces <li>Similarity <li>Nilpotence <li>Jordan Form <li>Topic: Geometry of Eigenvalues <li>Topic: The Method of Powers <li>Topic: Stable Populations <li>Topic: Linear Recurrences </ol>
 * 1) Factoring and Complex Numbers: A Review
 * 2) Complex Representations
 * 1) Definition and Examples
 * 2) Diagonalizability
 * 3) Eigenvalues and Eigenvectors
 * 1) Self-Composition
 * 2) Strings
 * 1) Polynomials of Maps and Matrices
 * 2) Jordan Canonical Form

Unitary Transformations
<ol type="I"> <li>Inner product spaces <li>Unitary and Hermitian matrices <li>Singular Value Decomposition <li>Spectral Theorem </ol>

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

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