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In this chapter, we briefly discuss some topics that are needed for the sequel. This chapter should be skimmed quickly and used primarily as a reference.

Let us begin with the definition of one of our principal objects of study.

Loosely speaking, a linear transformation is a function from one vector space to another that the vector space operations. Let us be more precise.

Let be a subspace of a vector space . It is easy to see that the binary relation on defined by is an equivalence relation. When ≡ , we say that and are . The term is used as a colloquialism for modulo and ≡ is often written When the subspace in question is clear, we will simply write ≡ .

Let be a vector space over a field and let . Then for any polynomial () ∈ [], the operator () is well-defined. For instance, if () = 1 + 2 + then () = + 2 + where is the identity operator and is the threefold composition ○ ○ .

Since all bases for a vector space have the same cardinality, the concept of vector space dimension is well-defined. A similar statement holds for free -modules when the base ring is commutative (but not otherwise).

We remind the reader of a few of the basic properties of principal ideal domains.

In this chapter, we study the structure of a linear operator on a finite-dimensional vector space, using the powerful module decomposition theorems of the previous chapter. .

Unless otherwise noted, we will assume throughout this chapter that all vector spaces are finite-dimensional.

We now turn to a discussion of real and complex vector spaces that have an additional function defined on them, called an , as described in the upcoming definition. Thus, in this chapter, will denote either the real or complex field. If is a complex number then the complex conjugate of is denoted by .