 Cyclic vector

For a linear operator $T$, a cyclic vector is a vector whose iterates span the entire vector space. In finite dimension $d$, the vectors $v,Tv,\dots ,T^{d-1}v$ then form a basis: the first linear dependence makes the preceding span invariant under $T$, so later iterates cannot enlarge it. In topological settings one often requires the iterates to have dense span instead; that closure convention must be stated.

In the context of linear algebra and vector spaces, a cyclic vector is a vector that generates a cyclic subspace under the action of a linear operator.