Mathematical Preliminaries

Abstract

The primary purpose of this chapter is to introduce a number of preliminary mathematical concepts which are fundamental to the remainder of the text. We assume that the reader is familiar with certain basic notions in linear algebra, especially those involving matrix terminology and manipulation, but that he does not wish to consult an outside source in order to familiarize himself with certain more abstract concepts which will be required. We begin by formally defining and illustrating the concept of a “linear vector space v over a field F” in Section 2.2. In this way, we are then able to introduce the notions of “linearly independent (or dependent) vectors”, a “basis” for an n-dimensional vector space, and the use of scalar matrices to represent a change of basis. In Section 2.3, we introduce “linear operators” and illustrate the use of scalar matrices to represent finite dimensional linear operators in a given basis as well as an altered basis. A number of important and useful definitions and results involving scalar matrices are then presented in Section 2.4. In particular, “matrix rank” is formally defined, and various tests for rank determination are outlined.