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# 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 $x,T(x),\ldots,T^{n}(x),\ldots\in Z(x,T)$, we have that

$W:=\operatorname{span}\{x,T(x),\ldots,T^{n}(x),\ldots\}\subseteq Z(x,T).$ |

On the other hand, since $W$ is $T$-invariant, $Z(x,T)\subseteq W$. Hence $Z(x,T)$ is the subspace generated by $x,T(x),\ldots,T^{n}(x),\ldots$ 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 $T$.

Defines:

cyclic vector

Related:

CyclicDecompositionTheorem, CyclicVectorTheorem

Synonym:

cyclic vector subspace

Major Section:

Reference

Type of Math Object:

Definition

Parent:

## Mathematics Subject Classification

15A04*no label found*47A16

*no label found*

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