Syllabus
Matrices, Geometry&Mathematica
Authors:  Bill Davis and Jerry Uhl  ©1999
Producer:  Bruce Carpenter
Publish er:         Distributor:  

These lessons are used for Math 415 and for Math 225

MGM.00  PlotFest

Using Mathematica to plot in two and three dimensions with special attention to parametric plotting.

MGM.01  Perpendicular Frames in 2D and 3D

Vectors in 2D and vectors in 3D.  Addition and subtraction of vectors.  Dot product and Cross product.
Aligning and hanging on perpendicular fram es to plot tilted ellipses and ellipsoids.  Right hand frames versus left hand frames.  Resolution of vectors into perpendicular components. &n bsp;Planes and lines through the origin.

MGM.02  2D Matrix Action
     

Matrix multiplication. Hitting the unit circle with a matrix and observing the result through matrix action movies. Linearity of matrix multiplication.  Takin g a 2D perpendicular frame and using it to to plot tilted ellipses.  Rotation matrices and right hand frames.  Reflection matrices and left hand fra mes.  Stretcher matrices.  Why  is unlikely to be the same as  for given 2D matrices  and < IMG SRC="Images/315syl_gr_10.gif" BORDER="0" ALT="[Graphics:Images/315syl_gr_10.gif]" WIDTH="11" HEIGHT="14" ALIGN="absmiddle" >.  Inverse matrices.

MGM.03  Making 2D Matrices
     

Using two prependicular frames and two stretch factors to make matrices whos hits have desired outcomes.  Inverting matrices made this way.  Making matrices whose hits stretch along a given perpndicular frame, making matrices whose hits reflect about a given line, making matrices whoses hits project onto a given li ne.  Ray tracing.  Parabolic, spherical, elliptic and hyperbolic reflectors, stealth technology.

MGM.04  SVD Analysis of 2D Matrices
   

The SVD (Singular Value Decomposition) says that corresponding to any 2D matrix  are two perpendicular frames and two stretch factors that can be used to duplicate .  Using SVD stretch factors to recognize invertible matrics and then invert them.  The determinant of a 2D matrix in terms of the SVD stretch factors.  Why the determinant of  is the inverse of the determinant of .  Rank of a 2D matrix.  Using 2D matrices to so lve systems of linear equations. Eigenvalues and eigenvectors of 2D matrices.
Optional: Hand calculations involving Cramer's rule and Gaussian elimination.

MGM.05  3D Matrices

This lesson repeats the ideas of MGM.02, MGM.03 and MGM.04 in 3D.

MGM.06  Beyond 3D
       

The SVD (Singular Value Decomposition) says that corresponding to any arbitray matrix  (square or non-square) are two perpendicular frames and a list of stretch factors that can be used to duplicate .  Rank of a matrix in terms of the SVD stretch factors.  The meaning of full rank.  Recognizing when a given system of n linear equations in  unknowns has:
a) exactly one solution (exactly determined).
b) many solutions (under determined)
c) no solution (over determined).
How to find find solutions of linear systems when they exist. 
Using SVD to explicitly cons truct the the PseudoInverse for getting best least squares solutions to over determined systems of linear equations.

MGM.07  Roundoff

Creative rounding of matrices via the Singular Value Decomposition and image compression.  Principal Component Analysis (PCA) of data via the Singular Value D ecomposition.  Ill-conditioned matrices: The trouble ill-conditioned matrices can cause and how to use the Singular Value Decomposition to recognize them.

MGM.08  Subspaces

Every set of vectors in nD spans a subspace of nD.  Projecting onto a subspace of nD.  Calculating the dimension of a subspace of nD.  A s et of  vectors in nD is linearly in dependent if it spans a -dimensiona l subspace of nD.  Traditional definitions of linear independence.  Orthonormal sets. Gram Schmidt process.  Alien plots coming from proje ctions of highD surfaces onto 3D subspaces.  Perpendicular complement of a subspace.  Null spaces of matrices.

MGM.09  Eigensense:Diagonalizable Matrices, 
Matrix Exponential, Matrix Powers and Dynamical Systems
      

Eigenvalues, eigenvectors and using them to recognize diagonalizable matrices.  Complex eigenvalues and eigenvectors.  The matrix exponential for di agonalizable and non-diagonalizable matrices.  Eigenvalues reveal long term behavior of matrix exponentials and matrix powers.  Using matrix exponen tials and matrix powers to solve continuous dynamical systems (systems of linear diffeerential equations) and discrete dynamical systems (systems of difference equation s).

MGM.10  The Spectral Theorem for Symmetric Matrices and the 
Holy Grail of Matrix Theory
         

This is the main theoretical lesson.  Discussion of the spectral theorem and its proof.  Given an arbitrary matrix , using the spectral theorem applied to  to explain why every matrix has a singular value decompositio n.
Using an orthonormal basis of eigenvectors of  to read off:
a) an orthonormal basis of the column space  of the matrix 
b) an orthonormal basis of the null space  of the matrix 
c) an  orthonormal basis of the row space of the matrix  
d) a construction of the PseudoInverse of the matrix 
Positive definite and positive semidefinite matrices.  Quadratic forms. &nbs p;Grammian matrices.

MGM.11  Function Spaces
         

Functions as vectors.  The root-mean-square distance between two functions on an interval.  Weighted root-mean-square distance.  The dot p roduct of two functions.  The component of one function in the direction of another.  Orthogonal sets of functions: Sine systems, Cosine systems, Si n-Cosine systems, Legendre Polynomial system.
Sets of functions orthogonal with respect to a weight function.  Chebyshev polynomials.  Gram-Schmi dt process.  Fourier approximation and orthogonal functions.  Fourier Sine approximation and the heat and wave equations.  Using Fourier m ethods to bring the Dirac Delta function to life.