Abstract algebra is not generally taught by examining its applications. Indeed, there is even a trend towards avoiding the intuitive and the applied and insisting on the abstract and axiomatic points of view. This might be right for mathematics majors, but is certainly wrong for future teachers. We propose an application-oriented approach in this chapter.

In section 8.1, a step by step, very detailed discussion of changes of temperature in a thin rod leads us to four-dimensional vectors, linear transformations, bases, eigenvalues and eigenvectors. The exercises are an integral part of the exposition and should be done very carefully.

The next section is written as a text at the eighth grade level. We stray here somewhat from our usual procedure, which is to introduce the mathematical notions as they develop while we examine the application. Instead, the application has been squarely based on the use of matrices. Still, it is an effective way of teaching matrices since cryptography is such an attractive subject. Matrix multiplication, inverses, and arithmetic modulo 26 are discussed at a basic level.

The natural algebra of linear differential operators is studied in section 8.3, up to the mathematical formulation of Heisenberg's uncertainty relation and the solution of some linear differential equations. Remarks for the teacher on how to introduce the algebraic aspects of calculus end this section.

Heat conduction II

The difference equation

We shall start with heat conduction in a thin insulated rod. We locate a point on the rod by its distance x from the left endpoint. The endpoints are labeled 0 and L, where L is the length of the rod.