ABSTRACT
In this chapter, the authors define what a vector space is, derive additional properties of a vector space implied by the axioms of the definition, and examine several examples. A vector will be an object in a vector space that will not necessarily have any geometric properties attached to it. A finite dimensional vector space is defined to be a vector space that has a finite spanning set. A vector space that is not finite dimensional is said to be infinite dimensional. (Showing that an infinite dimensional vector spaces has a basis is proven using Zorn’s lemma or a logically equivalent axiom, which is beyond the scope of this text.) Isomorphic vector spaces are structurally the same; the only real difference is how the elements are named. The specific theorem gives a succinct categorization of the finite dimensional vector spaces.
