Vector Spaces and Linear Equations

Abstract

The reader will easily believe that much of the difficulty of group theory may be attributed to the fact that the group operation is not assumed to be commutative. Indeed we mentioned in 18.9.9 towards the end of Chapter 18 a result on finitely generated abelian groups, which gives a complete statement of the algebraic structure of such groups. No such statement is known at present for arbitrary finitely-generated groups, and the problems involved in analysing their structure are indeed formidable. In this chapter we discuss a simpler theory even than that of commutative groups, namely vector spaces. However, it is not just in the interests of simplicity that we introduce the notion of a vector space. It is rather because vector spaces play a decisive role in Euclidean geometry and the theory of linear equations that they figure so prominently in modern mathematics courses.