*Linear Algebra*

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*Linear Algebra*

*An Introduction to Mathematical Discourse*

This book requires that you are familiar with calculus. This subject is covered by the wikibook Calculus. |

This book discusses *proof-based* linear algebra. 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.

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:**

### 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

### Vector Spaces

- Definition of Vector Space
- Linear Independence
- Basis and Dimension
- Topic: Fields
- Topic: Crystals
- Topic: Voting Paradoxes
- Topic: Dimensional Analysis

### Maps Between Spaces

- Isomorphisms
- Homomorphisms
- Computing Linear Maps
- Matrix Operations
- Change of Basis
- Projection
- Topic: Line of Best Fit
- Topic: Geometry of Linear Maps
- Topic: Markov Chains
- Topic: Orthonormal Matrices

### Determinants

- Definition
- Geometry of Determinants
- Other Formulas for Determinants
- Topic: Cramer's Rule
- Topic: Speed of Calculating Determinants
- Topic: Projective Geometry

### Similarity

- Complex Vector Spaces
- Similarity
- Nilpotence
- Jordan Form
- Topic: Geometry of Eigenvalues
- Topic: The Method of Powers
- Topic: Stable Populations
- Topic: Linear Recurrences

### Unitary Transformations

### Appendix

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

### Resources and Licensing

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