Linear Algebra

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Linear Algebra
An Introduction to Mathematical Discourse

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.

Table of Contents

This book is part of a series on Algebra:

Basic Algebra
Algebra
Intermediate Algebra
Linear Algebra
Abstract Algebra
Intro to Rings and Algebras

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)

Appendix

Resources And Licensing