cyclic subspace

Start with a single vector v and keep hitting it with T: you get v, Tv, T^2 v, T^3 v, and so on. The span of this whole orbit is the cyclic subspace generated by v. It is the smallest invariant subspace that contains v — you cannot have v inside an invariant subspace without dragging in all its T-images too.

Write it as Z(v, T) = span{ v, Tv, T^2 v, ... }. Because V is finite-dimensional, the powers eventually become linearly dependent: there is a smallest k with T^k v a combination of v, Tv, ..., T^(k-1) v. Then v, Tv, ..., T^(k-1) v form a basis of Z(v, T), and that polynomial relation is the minimal polynomial of T restricted to this subspace.

In the basis v, Tv, ..., T^(k-1) v the restriction of T looks like a companion matrix, the building block of rational canonical form. So cyclic subspaces are how you decompose an operator without ever leaving the base field — no eigenvalues, no field extension required. The catch: an operator is cyclic on all of V only when its minimal and characteristic polynomials coincide.