NetMath at the University of Illinois

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

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 frames to plot tilted ellipses and ellipsoids.  Right hand frames versus left hand frames.  Resolution of vectors into perpendicular components.  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.  Taking a 2D perpendicular frame and using it to to plot tilted ellipses.  Rotation matrices and right hand frames.  Reflection matrices and left hand frames.  Stretcher matrices.  Why A.B is unlikely to be the same as B.A for given 2D matrices A and B.  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 line.  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 A are two perpendicular frames and two stretch factors that can be used to duplicate A.  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 Inverse[A] is the inverse of the determinant of A.  Rank of a 2D matrix.  Using 2D matrices to solve 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 A (square or non-square) are two perpendicular frames and a list of stretch factors that can be used to duplicate A.  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 k 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 construct the the PseudoInverse for getting best least squares solutions to over determined systems of linear equations.

MGM.07  Roundoff (Optional)

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

Created by Mathematica  (September 6, 2006)

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