## You are here

Homecyclic subspace

## Primary tabs

# 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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

## Attached Articles

## Corrections

also defines cyclic vector by mathcam ✓

Strange notation by Johan ✓

appearance by CWoo ✓

classification by CWoo ✓

Strange notation by Johan ✓

appearance by CWoo ✓

classification by CWoo ✓