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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 has a collection for ordering a printed copy of this book for a fee. (edit) (help)

A printable version of Linear Algebra is available in two parts. Part 1edit contains everything from the beginning of the book through Chapter II. Part 2edit contains everything from Chapter III through the end of the book.

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:

Linear Systems

  1. Solving Linear Systems100% developed  as of Jul 13, 2009 (Jul 13, 2009)
    1. Gauss' Method 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
    2. Describing the Solution Set 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
    3. General = Particular + Homogeneous 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
    4. Comparing Set Descriptions 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
    5. Automation 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
  2. Linear Geometry of n-Space 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
    1. Vectors in Space 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
    2. Length and Angle Measures 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
  3. Reduced Echelon Form 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
    1. Gauss-Jordan Reduction 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
    2. Row Equivalence 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
  4. Topic: Computer Algebra Systems 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
  5. Topic: Input-Output Analysis 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
  6. Input-Output Analysis M File 100% developed  as of Mar 24 2008 (Mar 24 2008)
  7. Topic: Accuracy of Computations 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
  8. Topic: Analyzing Networks 100% developed  as of Jul 13, 2009 (Jul 13, 2009)
  9. Topic: Speed of Gauss' Method 50% developed  as of Mar 24, 2008 (Mar 24, 2008)

Vector Spaces 100% developed  as of Apr 17, 2009 (Apr 17, 2009)

  1. Definition of Vector Space100% developed  as of Apr 17, 2009 (Apr 17, 2009)
    1. Definition and Examples100% developed  as of Jun 18, 2009 (Jun 18, 2009)
    2. Subspaces and Spanning sets100% developed  as of Jun 18, 2009 (Jun 18, 2009)
  2. Linear Independence100% developed  as of Apr 17, 2009 (Apr 17, 2009)
    1. Definition and Examples100% developed  as of Apr 17, 2009 (Apr 17, 2009)
  3. Basis and Dimension100% developed  as of Apr 17, 2009 (Apr 17, 2009)
    1. Basis100% developed  as of Jun 18, 2009 (Jun 18, 2009)
    2. Dimension100% developed  as of Apr 17, 2009 (Apr 17, 2009)
    3. Vector Spaces and Linear Systems100% developed  as of Apr 17, 2009 (Apr 17, 2009)
    4. Combining Subspaces100% developed  as of Apr 17, 2009 (Apr 17, 2009)
  4. Topic: Fields100% developed  as of Apr 17, 2009 (Apr 17, 2009)
  5. Topic: Crystals100% developed  as of Apr 17, 2009 (Apr 17, 2009)
  6. Topic: Voting Paradoxes100% developed  as of Apr 17, 2009 (Apr 17, 2009)
  7. Topic: Dimensional Analysis100% developed  as of Apr 17, 2009 (Apr 17, 2009)

Maps Between Spaces

  1. Isomorphisms100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    1. Definition and Examples100% developed  as of July 19, 2009 (July 19, 2009)
    2. Dimension Characterizes Isomorphism100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  2. Homomorphisms100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    1. Definition of Homomorphism100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    2. Rangespace and Nullspace100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  3. Computing Linear Maps100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    1. Representing Linear Maps with Matrices100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    2. Any Matrix Represents a Linear Map100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  4. Matrix Operations100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    1. Sums and Scalar Products100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    2. Matrix Multiplication100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    3. Mechanics of Matrix Multiplication100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    4. Inverses100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  5. Change of Basis100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    1. Changing Representations of Vectors100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    2. Changing Map Representations100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  6. Projection100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    1. Orthogonal Projection Onto a Line100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    2. Gram-Schmidt Orthogonalization100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    3. Projection Onto a Subspace100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  7. Topic: Line of Best Fit100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  8. Topic: Geometry of Linear Maps100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  9. Topic: Markov Chains100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  10. Topic: Orthonormal Matrices100% developed  as of Jun 21, 2009 (Jun 21, 2009)

Determinants100% developed  as of Jun 21, 2009 (Jun 21, 2009)

  1. Definition100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    1. Exploration100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    2. Properties of Determinants100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    3. The Permutation Expansion100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    4. Determinants Exist100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  2. Geometry of Determinants100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    1. Determinants as Size Functions100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  3. Other Formulas for Determinants100% developed  as of Jun 21, 2009 (Jun 21, 2009)
    1. Laplace's Expansion100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  4. Topic: Cramer's Rule100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  5. Topic: Speed of Calculating Determinants100% developed  as of Jun 21, 2009 (Jun 21, 2009)
  6. Topic: Projective Geometry100% developed  as of Jun 21, 2009 (Jun 21, 2009)

Similarity100% developed  as of Jun 24, 2009 (Jun 24, 2009)

  1. Complex Vector Spaces100% developed  as of Jun 24, 2009 (Jun 24, 2009)
    1. Factoring and Complex Numbers: A Review100% developed  as of Jun 24, 2009 (Jun 24, 2009)
    2. Complex Representations100% developed  as of Jun 24, 2009 (Jun 24, 2009)
  2. Similarity
    1. Definition and Examples100% developed  as of Jun 24, 2009 (Jun 24, 2009)
    2. Diagonalizability100% developed  as of Jun 24, 2009 (Jun 24, 2009)
    3. Eigenvalues and Eigenvectors100% developed  as of Jun 24, 2009 (Jun 24, 2009)
  3. Nilpotence100% developed  as of Jun 24, 2009 (Jun 24, 2009)
    1. Self-Composition100% developed  as of Jun 24, 2009 (Jun 24, 2009)
    2. Strings100% developed  as of Jun 24, 2009 (Jun 24, 2009)
  4. Jordan Form100% developed  as of Jun 24, 2009 (Jun 24, 2009)
    1. Polynomials of Maps and Matrices100% developed  as of Jun 24, 2009 (Jun 24, 2009)
    2. Jordan Canonical Form100% developed  as of Jun 24, 2009 (Jun 24, 2009)
  5. Topic: Geometry of Eigenvalues50% developed  as of Jun 24, 2009 (Jun 24, 2009)
  6. Topic: The Method of Powers100% developed  as of Jun 24, 2009 (Jun 24, 2009)
  7. Topic: Stable Populations100% developed  as of Jun 24, 2009 (Jun 24, 2009)
  8. Topic: Linear Recurrences100% developed  as of Jun 24, 2009 (Jun 24, 2009)

Unitary Transformations

  1. Inner product spaces75% developed
  2. Unitary and Hermitian matrices75% developed
  3. Singular Value Decomposition75% developed
  4. Spectral Theorem75% developed

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

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