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 for an operator and introduces the concept of 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.
