ABSTRACT
This chapter introduces the most important number systems culminating with the complex numbers. It also introduces multiplication of matrices and explores some properties of matrices analogous to the number systems. The chapter explores two applications: least squares fitting of data and changing plane coordinates. The fact that each of these relies on matrix arithmetic and symmetric projectors is a good example of the paradox that the great utility of mathematics derives from its apparent abstraction. By expanding the use of addition and multiplication beyond ordinary numbers, matrices open the door to “abstract algebra.” Matrix arithmetic vastly expands the “mathematical tool kit,” and specifically “representation theory” uses matrix arithmetic as a tool to understand more subtle mathematical structures. The chapter describes block matrices and utilizes them with remainder matrices to obtain a very famous matrix factorization of certain Vandermonde matrices called the “fast Fourier transform”.
