Linear Algebra

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

Linear Systems

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

Vector Spaces

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

Maps Between Spaces

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

Determinants

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

Similarity

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

Appendix

Resources And Licensing

This article is issued from Wikibooks. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.