ABSTRACT
The most basic object in linear algebra is that of a vector space. Vector spaces arise in nearly every possible mathematical context and often in concrete ones as well. In this chapter, we develop the fundamental concepts for describing and characterizing vectors spaces. In the first section, we define and enumerate the properties of fields. Examples are the rational numbers, the reals, and the complex numbers. Basically, a field is determined by those properties necessary to solve all systems of linear equations. The second section is concerned with the space Fn, where n is a natural number and F is any field. These spaces resemble the real vector space Rn and the complex space Cn. In section three, we introduce the abstract concept of a vector space, as well as subspace and give several examples. The fourth section is devoted to the study of subspaces of a vector space V, and we establish a criteria for a subset to be a subspace that substantially reduces the number of axioms which have to be demonstrated. In section five, we introduce the concepts of linear independence and Span. Section six deals with bases and dimension in finitely generated vector spaces. In section seven, we prove that every vector space has a basis. In the final section, we show, given a basis for an n-dimensional vector space V over a field F, how to associate a vector in Fn. This is used to translate questions of independence and spanning in V to the execution of standard algorithms in Fn.
