cyclic subspace

Let $V$ be a vector space over a field $k$, and $x\in V$. Let $T:V\to V$ be a linear transformation. The $T$-cyclic subspace generated by $x$ is the smallest $T$-invariant subspace which contains $x$, and is denoted by $Z(x,T)$.

Since $x,T(x),\mathrm{\dots },T^n(x),\mathrm{\dots }\in Z(x,T)$, we have that

$$W:=\mathrm{span}\{x,T(x),\mathrm{\dots },T^n(x),\mathrm{\dots }\}\subseteq Z(x,T).$$

On the other hand, since $W$ is $T$-invariant, $Z(x,T)\subseteq W$. Hence $Z(x,T)$ is the subspace generated by $x,T(x),\mathrm{\dots },T^n(x),\mathrm{\dots }$ In other words, $Z(x,T)=\{p(T)(x)\mid p\in k[X]\}$.

Remark. If $Z(x,T)=V$ we say that $x$ is a cyclic vector of $T$.