Chapter 1 Fields and Matrices

3cm My view is that mathematics is primarily a language for modeling the physical world, or various abstractions of the physical world.

– Terry Tao (1975 - )

Fields medallist

We will begin by introducing a generalization of the real numbers (i.e. fields), and then restating some facts, notation, and techniques from MATH105 using the language of fields. You are already expected to be familiar with row reduction, row operations, inverse matrices, upper and lower triangular matrices, at least for matrices of real numbers, so we will not spend much time reintroducing those.

A field, as defined below, is an abstract mathematical structure. The most commonly used examples of fields are the rational numbers $\mathbb{Q}$, the real numbers $ℝ$, and the complex numbers $ℂ$. We will discuss some other examples as well. The purpose for making this abstract definition is that much of the theory of linear algebra is still valid, regardless of one’s choice of field.