Three - Vector Spaces and Linear Transformations

Abstract

This chapter provides an overview on vector spaces and linear transformations. The similarities between the set of vectors in Rn and the set of functions arise not so much because of the nature of the elements of the sets but because of the nature of the operations defined on the sets. It is, therefore, more efficient to consider, in an abstract way, sets of objects whose elements are not defined but on which are defined operations of multiplication by a number and addition. Sets possessing these properties are called vector spaces or linear spaces. The chapter presents the examples of vector spaces whose elements are numbers, matrices, infinite sequences, and other mathematical objects. In advanced treatments of linear algebra, vector spaces over arbitrary fields are considered. The elements of a field are called as scalars.