ABSTRACT
The central notions in linear algebra are vector spaces and linear transformations that act between vector spaces. We will define these notions in Chapters 2 and 3, respectively. But before we can introduce the general notion of a vector space we need to talk about the notion of a field. In your first Linear Algebra course you probably did not worry about fields because it was chosen to only talk about the real numbers R, a field you have been familiar with for a long time. In this chapter we ask you to get used to the general notion of a field, which is a set of mathematical objects on which you can define algebraic operations such as addition, subtraction, multiplication and division with all the rules that also hold for real numbers (commutativity, associativity, distributivity, existence of an additive neutral element, existence of an additive inverse, existence of a multiplicative neutral element, existence of a multiplicative inverse for nonzeros). We start with an example.
