The standard approach to eigenvalues and eigenvectors in texts is to demonstrate algebraic techniques for finding the eigenvalues and eigenvectors of a matrix A, which then allows A to be "diagonalized" as A = PDP^-1. We will instead use what we know about the geometry of a simple transformation in two dimensions to determine the eigenvalues and eigenvectors of the transformation, and from them we will construct the matrix of the transformation. After doing this, we will analyze the product PDP^-1 for our transformation. This analysis will reveal the richness of this simple example; we'll see a variety of fundamental ideas from a first course in linear algebra.

This study incorporates the ideas of complexity and chaos and is based on a series of daily surveys of 254 undergraduate engineering majors enrolled in 10 intensive mathematics classes over 73 days during the entire semester. The findings suggest a hypothesis that patterns of chaos in the dynamics of student attitudes toward mathematics class might represent not only random noise or measurement error but may develop through series of bifurcations associated with such educational parameters as SAT/ACT math score, GPA, course grade, and gender.

Do you enjoy problems that seem impossible and turn out to be simple? How about the ones that seem easy but are devilishly challenging? I'll share some of my favorites, including one that took me over 25 years to solve!