Discussion of Tools for Advanced Linear Algebra

One of the driving problems of advanced linear algebra, as encouraged by Strang (1988, p. 197), is the discussion of the singular value decomposition, for "the SVD has become fundamental in scientific computing." In particular, the SVD solves the problem of diagonalizing a matrix, which differs from an algebra-oriented course that might use the Jordan canonical form to answer that question.

By considering the question

What is the shape of the image of the unit circle under the transformation T(x) = Ax ?

we can nudge students toward thinking about the geometry associated with the transformation, rather than relying entirely on computations.

For instance, if we consider the matrix

A = ,

we see this answer:

These pictures provoke two questions:

1. Is the image of the circle under the transformation T really an ellipse?
2. If it is an ellipse, what are the axes?

In the early years, Tom Hern and the late Cliff Long used hand-drawn pictures to motivate a discussion. Then, over time, technology allowed them to portray these pictures in a more professional manner -- but still static in nature. Finally, they used MATLAB to draw the unit circle, and, as the unit circle was traced out, the image curve would be traced. Now, with the advent of JavaSketchpad , the static pictures have become dynamic, web-based tools, namely WebSVD and Hern & Long SVD, allowing students to explore the singular value decomposition and explain the components of the SVD.

Next  or  or  page: 17. WebSVD Tool and Sample Activity