Linear Transformations

Abstract

As shown in Chapter 3, there are many different vector spaces even with the same dimension . The question now is how one can determine whether or not two given vector spaces have the ‘same’ structure as vector spaces, or can be identified as the same vector space. To answer the question , one has to compare them first as sets, and then see whether their arithmetic rules are the same or not. A usual way of comparing two sets is to define a function between them. When a function f is given between the underlying sets of vector spaces, one can compare the arithmetic rules of the vector spaces by examining whether the function f preserves two algebraic operations: the vector addition and the scalar multiplic ation, that is, f(x + y) = f(x) + f(y) and f(kx) = kf(x) for any vectors x, y and any scalar k. In this chapter, we discuss this kind of functions between vector spaces.