Linear Algebra

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This book requires that you first read Algebra.
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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

[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

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[edit] Linear Systems

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

[edit] Vector Spaces

Vector SpacesDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
  1. Definition of Vector SpaceDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
    1. Definition and ExamplesDevelopment stage: 100% (as of Jun 18, 2009)(Jun 18, 2009)
    2. Subspaces and Spanning setsDevelopment stage: 100% (as of Jun 18, 2009)(Jun 18, 2009)
  2. Linear IndependenceDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
    1. Definition and ExamplesDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
  3. Basis and DimensionDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
    1. BasisDevelopment stage: 100% (as of Jun 18, 2009)(Jun 18, 2009)
    2. DimensionDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
    3. Vector Spaces and Linear SystemsDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
    4. Combining SubspacesDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
  4. Topic: FieldsDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
  5. Topic: CrystalsDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
  6. Topic: Voting ParadoxesDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)
  7. Topic: Dimensional AnalysisDevelopment stage: 100% (as of Apr 17, 2009)(Apr 17, 2009)

[edit] Maps Between Spaces

  1. IsomorphismsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    1. Definition and ExamplesDevelopment stage: 100% (as of July 19, 2009)(July 19, 2009)
    2. Dimension Characterizes IsomorphismDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  2. HomomorphismsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    1. Definition of HomomorphismDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    2. Rangespace and NullspaceDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  3. Computing Linear MapsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    1. Representing Linear Maps with MatricesDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    2. Any Matrix Represents a Linear MapDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  4. Matrix OperationsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    1. Sums and Scalar ProductsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    2. Matrix MultiplicationDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    3. Mechanics of Matrix MultiplicationDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    4. InversesDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  5. Change of BasisDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    1. Changing Representations of VectorsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    2. Changing Map RepresentationsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  6. ProjectionDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    1. Orthogonal Projection Into a LineDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    2. Gram-Schmidt OrthogonalizationDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    3. Projection Into a SubspaceDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  7. Topic: Line of Best FitDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  8. Topic: Geometry of Linear MapsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  9. Topic: Markov ChainsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  10. Topic: Orthonormal MatricesDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)

[edit] Determinants

DeterminantsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  1. DefinitionDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    1. ExplorationDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    2. Properties of DeterminantsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    3. The Permutation ExpansionDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    4. Determinants ExistDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  2. Geometry of DeterminantsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    1. Determinants as Size FunctionsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  3. Other Formulas for DeterminantsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
    1. Laplace's ExpansionDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  4. Topic: Cramer's RuleDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  5. Topic: Speed of Calculating DeterminantsDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)
  6. Topic: Projective GeometryDevelopment stage: 100% (as of Jun 21, 2009)(Jun 21, 2009)

[edit] Similarity

Introduction to SimilarityDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
  1. Complex Vector SpacesDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
    1. Factoring and Complex Numbers; A ReviewDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
    2. Complex RepresentationsDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
  2. SimilarityDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
    1. DiagonalizabilityDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
    2. Eigenvalues and EigenvectorsDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
  3. NilpotenceDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
    1. Self-CompositionDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
    2. StringsDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
  4. Jordan FormDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
    1. Polynomials of Maps and MatricesDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
    2. Jordan Canonical FormDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
  5. Topic: Geometry of EigenvaluesDevelopment stage: 50% (as of Jun 24, 2009)(Jun 24, 2009)
  6. Topic: The Method of PowersDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
  7. Topic: Stable PopulationsDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)
  8. Topic: Linear RecurrencesDevelopment stage: 100% (as of Jun 24, 2009)(Jun 24, 2009)

[edit] Appendix

[edit] Resources And Licensing

[edit] Old pages to be implemented

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[edit] External links

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