Linear Algebra

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A Wikibookian has nominated this book for cleanup because: This "book" consists of 2 parts: Fragmented pages from the old book, and a recently donated book. This book needs to be cleaned up. You can help make it better. Please review any relevant discussion.

This book requires that you first read Algebra.

This book is intended for advanced readers.

Linear algebra is a branch of algebra in mathematics concerned with the study of vectors, vector spaces, linear transformations, and systems of linear equations. Vector spaces are very important in modern mathematics. Linear algebra is widely used in abstract algebra and functional analysis. It has extensive applications in natural and social sciences, for both linear systems and linear models of nonlinear systems.

It is part of the study of Abstract Algebra.

Contents 1 General Information and MoS 1.1 Exercises 2 Table of Contents 2.1 Linear Systems 2.2 Vector Spaces 2.3 Maps Between Spaces 2.4 Determinants 2.5 Similarity 2.6 Appendix 2.7 Resources And Licensing 3 Old pages to be implemented 4 External links

[edit] General Information and MoS

This book is meant for students who wish to study linear algebra from scratch. The approach will not be entirely informal. Every result in the book is intended to be either proved or justified by some mathematical procedure. Links to tedious proofs can be made to Famous Theorems of Mathematics/Algebra after the proof is written there.

[edit] Exercises

Learning to think is extremely important in mathematics. Therefore in this book exercises form an important component and by no means should be ignored. Many important concepts of linear algebra are developed via the exercises in the book. It is necessary that before proceeding to the next chapter, the student does the exercises. Links to hints and solutions to many of the exercises are provided but they should be only used in cases of difficulty.

[edit] Table of Contents

A donated book has been uploaded here. This book was originally written by Jim Hefferon, and is available from this url: URL. The original text of the book is released under the terms of the GFDL and CC-BY-SA-2.5 licenses. The donated book is however not suitably formatted for wikibooks yet. Please assist in its formatting. For an example of what a page should look like after formatting please see the Gauss' Method in the book. See User:Whiteknight or User:Shahab for more information about this project.

• Cover
• Notation
• Introduction

[edit] Linear Systems

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

[edit] Vector Spaces

Vector Spaces (Apr 17, 2009)

1. Definition of Vector Space (Apr 17, 2009)
   1. Definition and Examples (Jun 18, 2009)
   2. Subspaces and Spanning sets (Jun 18, 2009)
2. Linear Independence (Apr 17, 2009)
   1. Definition and Examples (Apr 17, 2009)
3. Basis and Dimension (Apr 17, 2009)
   1. Basis (Jun 18, 2009)
   2. Dimension (Apr 17, 2009)
   3. Vector Spaces and Linear Systems (Apr 17, 2009)
   4. Combining Subspaces (Apr 17, 2009)
4. Topic: Fields (Apr 17, 2009)
5. Topic: Crystals (Apr 17, 2009)
6. Topic: Voting Paradoxes (Apr 17, 2009)
7. Topic: Dimensional Analysis (Apr 17, 2009)

[edit] Maps Between Spaces

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

[edit] Determinants

Determinants (Jun 21, 2009)

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

[edit] Similarity

Introduction to Similarity (Jun 24, 2009)

1. Complex Vector Spaces (Jun 24, 2009)
   1. Factoring and Complex Numbers; A Review (Jun 24, 2009)
   2. Complex Representations (Jun 24, 2009)
2. Similarity (Jun 24, 2009)
   1. Diagonalizability (Jun 24, 2009)
   2. Eigenvalues and Eigenvectors (Jun 24, 2009)
3. Nilpotence (Jun 24, 2009)
   1. Self-Composition (Jun 24, 2009)
   2. Strings (Jun 24, 2009)
4. Jordan Form (Jun 24, 2009)
   1. Polynomials of Maps and Matrices (Jun 24, 2009)
   2. Jordan Canonical Form (Jun 24, 2009)
5. Topic: Geometry of Eigenvalues (Jun 24, 2009)
6. Topic: The Method of Powers (Jun 24, 2009)
7. Topic: Stable Populations (Jun 24, 2009)
8. Topic: Linear Recurrences (Jun 24, 2009)

[edit] Appendix

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

[edit] Resources And Licensing

• Licensing And History
• Resources
• Bibliography
• Index (Jun 24, 2009)

[edit] Old pages to be implemented

A Wikibookian has nominated this book for cleanup because: These sections are left over remnants from other book projects, and need to be incorporated into the pages above. You can help make it better. Please review any relevant discussion.

• Laplace's Theorem
• Cofactors and Minors
• Fields
• Systems of Linear Equations  (Mar 12, 2008)
• Null spaces
• Invariant Subspaces
• Eigenvectors and Eigenvalues
• Linear transformations
• Matrices
• Row and column spaces
• Homogeneous Systems
• General Systems
• General Solutions
• Vector Spaces
• Linear Dependence
• Bases and dimensions
• Subspaces
• Direct Sum
• Quotient Space
• Span of a set
• Hyperplanes
• Linear Dependence
• Matrices and Determinants
• Cofactors and Minors
• Cramer's Rule
• Null Spaces (May 25, 2007)
• Column and Row Spaces (May 25, 2007)
• Systems of Linear Equations  (Oct 9, 2006)
• Row Reduction and Echelon Forms  (Apr 13, 2007)
• The Matrix Equation Ax=b
• Inner Product Spaces  (May 11, 2007)
• Matrices and Vectors
• Eigenvalues and Eigenvectors
• Zero Matrices and Zero Vectors  (Apr 13, 2007)
• Characteristic Equation
• Matrix Operations  (Jun 27, 2007)
• Cramer's Rule  (Oct 20, 2006)
• Change of Basis (Jun 27, 2007)
• Introduction to Determinants  (May 11, 2007)
• The Inverse of a Matrix  (May 11, 2007)
• Partitioned Matrices  (Apr 13, 2007)
• Vector Spaces And Subspaces (Need attention as this is a page from the old book)
• Inner Product, Length, and Orthogonality (It is a stub)
• Orthogonal Sets
• Augmented Matrices  (Oct 17, 2006)
• Linear Transformations  (May 11, 2007)
• Basis Vectors  (Oct 20, 2006)
• Linear Transformations (should be looked through, and maybe rewritten as it is taken from the "old" book)  (Oct 20, 2006)
• Glossary
• Matrices
• Vectors
• Linear transformations
• Eigenvalues and eigenvectors
• Row and Column Operations
• OLD/Vector Spaces
• OLD/Matrix Operations
• OLD/Eigenvalues and Eigenvectors
• OLD/Change of Basis
• Matrix Inverses
• Determinant
• Addition, Multiplication, and Transpose

[edit] External links

• A course in linear algebra - It contains a free set of video lectures given at the Massachusetts Institute of Technology.
• A free book on linear algebra - by Jim Hefferon (St. Michael's College)
• A First Course in Linear Algebra - a free textbook by Rob Beezer (University of Puget Sound) GNU
• Lecture Notes on Linear Algebra
• A toolkit for linear algebra students - This page is designed to help a linear algebra student learn and practice a basic linear algebra procedure, such as Gauss-Jordan reduction, calculating the determinant, or checking for linear independence. By Przemyslaw Bogacki. Department of Mathematics and Statistics, Old Dominion university.