V as an F[x]-module

Here is the single trick that powers all of canonical-form theory. Start with a vector space V and a fixed operator T. We already know how to multiply vectors by scalars from F; now we declare that the variable x acts as T. Once x means apply T, any polynomial p(x) = a_n x^n + ... + a0 acts as p(T) = a_n T^n + ... + a0 I. Suddenly V is a module over the polynomial ring F[x].

Precisely, define the scalar multiplication p(x) . v := p(T)(v). All the module axioms hold automatically because polynomials in T add and multiply just like polynomials in x (they commute, they distribute). The operator T is no longer a separate object floating beside V — it has been absorbed into the scalars themselves.

Why bother? This dictionary converts every operator question into a module question, where the powerful structure theorem for modules over a PID applies (F[x] is a PID). A T-invariant subspace becomes simply a submodule; a cyclic subspace becomes a cyclic module; the minimal polynomial becomes the generator of the annihilator ideal. Canonical forms drop out as the module decomposition.

The one thing to keep straight: different operators T on the same V give genuinely different F[x]-module structures. The underlying additive group is identical, but the action of x differs, so module isomorphism of (V, T) and (V, S) is exactly similarity of T and S. The module remembers the operator.