1 - Vector Spaces

Abstract

An arbitrary field can be used in place of either the field of real numbers or that of complex numbers—for example, a wide variety of mathematical systems can be classified as vector spaces over some field. This chapter discusses the idea of a subspace of a vector space, the elementary algebra of subspaces, and ways to generate subspaces. A vector quantity can be represented by an arrow, that is, a directed line segment. The study of the properties of addition of arrows and the multiplication of an arrow by a scalar can be transformed from a geometric to analgebraic setting by the introduction of a coordinate system in the space at hand. The elements of a vector space over a field are called vectors. The definition of a vector space over a field focuses attention on a set whose elements are called vectors. A wide variety of concrete mathematical systems are vector spaces; therefore, the axiomatic approach to the study of vector spaces is an efficient one.