This is the summary of a presentation given at the Joint Mathematics Meetings, January 10-13, 1996, Orlando, Florida.

In recent years there has been an explosion of interest in the fields of chaotic dynamical systems and fractal geometry. These fields are connected by the fact that when chaotic behavior is present in a given dynamical system it often occurs on a fractal. In this paper we describe a module appropriate for the sophomore level linear algebra course introducing some of these ideas. In particular we discuss a computer lab which introduces students to the notion of fractals generated via iterated function systems.

The geometry of linear transformations is a topic that is easily shortchanged in the traditional linear algebra course. Using the software Fractal Attraction students can enter a linear transformation L of the plane as a 2 x 2 matrix, and some intial geometric shape B. In a separate window they then see the image of B under L. In this way students actively explore the relationship between the geometry of the linear transformations scaling, rotation, reflection, projection and shearing, and the entries of the corresponding 2 x 2 matrix.

This connection is further reinforced through the investigation of fractals generated via iterated function systems (IFS). In class we discuss such fractals, examples of which are Sierpinski's triangle, the von Koch snowflake curve and the fern of Barnsley. It turns out there is an intimate connection between the geometry of linear transformations and fractals generated by an IFS, which students investigate in a compuer lab homework assignment. Students are amazed to see the images produced in this way and they are also given the (all too rare) opportunity to see some very current mathematics. As a follow up we disuss Barnsley's Collage Theorem and its implications for image compression, as well as fractal dimension.

Through this module students enhance their understanding of the geometry of linear trransformations, see modern mathematics, and have fun; hence the connection between fractals and linear algebra is a natural one.

The article "Fractals In Linear Algebra" is to appear in The College Mathematics Journal, May 1996.

James A. Walsh
Department of Mathematics
Oberlin College
Oberlin, OH 44074
email: jimw@cs.oberlin.edu