Linear Algebra

From Wikibooks, the open-content textbooks collection

This book or module has been nominated 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. Please edit this module to improve it. See this module's talk page for discussion.

This book requires Algebra as a prerequisite.

This book is intended for readers at the Advanced level.

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 The Book of Mathematical Proofs/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 Page
• Notation
• Preface

[edit] Linear Systems

• Solving Linear Systems (Mar 12, 2008)
• Gauss' Method (Mar 13, 2008)
  • Describing the Solution Set (Mar 14, 2008)
  • General = Particular + Homogeneous (Mar 17, 2008)
  • Comparing Set Descriptions (Mar 21, 2008)
  • Automation (Mar 21, 2008)
• Linear Geometry of n-Space (Mar 21, 2008)
  • Vectors in Space (Mar 21, 2008)
  • Length and Angle Measures (Mar 21, 2008)
• Reduced Echelon Form (Mar 21, 2008)
  • Gauss-Jordan Reduction (Mar 21, 2008)
  • Row Equivalence (Mar 21, 2008)
• Topic: Computer Algebra Systems
• Topic: Input-Output Analysis
  • Input-Output Analysis M File (Mar 24 2008)
• Topic: Accuracy of Computations
• Topic: Analyzing Networks
• Topic: Speed of Gauss' Method (Mar 24, 2008)

[edit] Vector Spaces

• Definition of Vector Space (Mar 24, 2008)
• Linear Independence (Mar 24, 2008)
  • Definition and Examples (Apr 6, 2008)
• Basis and Dimension
  • Dimension
  • Vector Spaces and Linear Systems
  • Combining Subspaces
• Topic: Fields
• Topic: Crystals
• Topic: Voting Paradoxes
• Topic: Dimensional Analysis

[edit] Maps Between Spaces

• Isomorphisms
  • Dimension Characterizes Isomorphism
• Homomorphisms
  • Definition of Homomorphism
  • Rangespace and Nullspace
• Computing Linear Maps
  • Representing Linear Maps with Matrices
  • Any Matrix Represents a Linear Map
• Matrix Operations
  • Sums and Scalar Products
  • Matrix Multiplication
  • Mechanics of Matrix Multiplication
  • Inverses
• Change of Basis
  • Changing Representations of Vectors
  • Changing Map Representations
• Projection
  • Orthogonal Projection Into a Line
  • Gram-Schmidt Orthogonalization
  • Projection Into a Subspace
• Topic: Line of Best Fit
• Topic: Geometry of Linear Maps
• Topic: Markov Chains
• Topic: Orthonormal Matrices

[edit] Determinants

• Determinants
  • Definition of Determinant
  • Exploration
  • Properties of Determinants
  • The Permutation Expansion
  • Determinants Exist
• Geometry of Determinants
  • Determinants as Size Functions
• Other Formulas for Determinants
  • Laplace's Expansion
• Topic: Cramer's Rule
• Topic: Speed of Calculating Determinants
• Topic: Projective Geometry

[edit] Similarity

• Introduction to Similarity
• Complex Vector Spaces
  • Factoring and Complex Numbers; A Review
  • Complex Representations
• Similarity
  • Diagonalizability
  • Eigenvalues and Eigenvectors
• Nilpotence
  • Self-Composition
  • Strings
• Jordan Form
  • Polynomials of Maps and Matrices
  • Jordan Canonical Form
• Topic: Geometry of Eigenvalues
• Topic: The Method of Powers
• Topic: Stable Populations
• Topic: Linear Recurrences

[edit] Appendix

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

[edit] Resources And Licensing

• Licensing And History
• Resources
• Bibliography
• Index

[edit] Old pages to be implemented

This book or module has been nominated for cleanup because: These sections are left over remnants from other book projects, and need to be incorporated into the pages above. Please edit this module to improve it. See this module's talk page for discussion.

• Null spaces
• Invariant Subspaces
• Eigenvectors and Eigenvalues
• Linear transformations
• Matrices
• Row and column spaces
• Homogeneous Systems
• General Systems
• General Solutions
• Vector Spaces
• Linear Dependance
• Bases and dimensions
• Subspaces
• Direct Sum
• Quotient Space
• Span of a set
• Hyperplanes
• Linear Dependance
• Fields
• Systems of Linear Equations
• 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

[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.