ABSTRACT
In this chapter we determine the structure of a single linear operator on a finite-dimensional vector space. The first section deals with the concept of an invariant subspace of an operator and the annihilator of a vector with respect to an operator. In section two we introduce the notion of a cyclic operator and uncover its properties. Section three concerns maximal vectors, in particular, we show that such vectors exist. Section four develops the theory of indecomposable operators. In section five we obtain our main structure theorem. This is applied in section six where we are able to obtain nice matrix representations for the similarity class of an operator. In the final section we specialize and apply these results to operators on finite-dimensional real and complex vector spaces.
