ABSTRACT
It is typical in the study of algebra to begin with the definition of its basic objects and investigate their properties. Then it is customary to introduce maps (functions, transformations) between these objects that preserve the algebraic character of the object. The relevant types of maps when the objects are vector spaces are linear transformations. In this chapter, we introduce and begin to develop the theory of linear transformations between vector spaces. In the first section, we define the concept of a linear transformation from a vector space V to a vector space W and give examples. In the second section, we define the kernel of a linear transformation. We prove a criteria for a transformation to be injective (one-to-one) in terms of the kernel. In section three, we prove some fundamental theorems about linear transformations, referred to as isomorphism theorems. In section four we consider a linear transformation T from an n-dimensional vector space V to an m-dimensional vector space W and show how, using a fixed pair of bases from V and W, respectively, to obtain an m×n matrix M for the linear transformation. This is used to define addition and multiplication of matrices. In the fifth section, we introduce the notion of an algebra over a field F as well as an isomorphism of algebras. We will show that for a finite dimensional vector space V over a field F the space L(V, V ) of operators on V is an algebra over F. We will also introduce the space Mnn(F) of n× n matrices with entries in the field F and show that this is an algebra isomorphic to L(V, V ), where V has dimension n. In the final section, we investigate linear transformations that are bijective. We investigate the relationship between two matrices, which arise as the matrix of the same transformation but with respect to different bases. This gives rise to the notion of a change of basis matrix. Of particular importance is the situation where the transformation is an operator on a space V and motivates the definition of similar operators and matrices.
