Notes for Math 523

1.     Vectors and Vector Operations
  1.1   Vectors
  1.2   Vector Operations
  1.3   The Algebra of Vector Operations
  1.4   Multiplication of Vectors and Linear Functions
  1.5   Multiplication of Vectors by Matrices
  1.6   Multiplication of Matrices
  1.7   Vectors and Matrices in Mathematica
  1.8   Vector & Matrix Operations in MATLAB

2.     Linear Equations
  2.1   Gaussian Elimination without Row Exchanges
  2.2   Gaussian Elimination with Row Exchanges
  2.3   Number of Equations Different From Number of Unknowns
  2.4   Subspaces
  2.5   Linear Independence and Linear Dependence
  2.6   Bases and Dimension
  2.7   The Four Subspaces of a Matrix
  2.8   Electrical Networks 
  2.9   Electrical Networks Calculations in Mathematica 
  2.10  Electrical Networks Calculations in MATLAB 

3.     Orthogonality
  3.1   Inner Products and Orthogonality
  3.2   Least Squares and Orthogonal Projections
  3.3   Least Squares Curve Fitting
  3.4   Orthonormal Sets

4.     Determinants
  4.1   Uses of Determinants  
  4.2   Definition of Determinants
  4.3   Properties of Determinants
  4.4   Applications of Determinants
    
5.     Eigenvalues and Eigenvectors
  5.1   Eigenvalues and Eigenvectors
  5.2   Diagonalization & Matrix Powers  
  5.3   Difference Equations 
  5.4   Differential Equations 
  5.5   Complex Eigenvalues
  5.6   Differential Equations with Complex Eigenvalues  
  5.7   Symmetric Matrices
  5.8   Diagonalizable Matrices
  5.9   Generalized Eigenvectors

6.     Quadratic Functions
  6.1   Representing Quadratic Functions
  6.2   Quadratic Forms
  6.3   Positive Definite Quadratic Forms

  An Introduction to Mathematica

  Mathematics 523 Directory