# 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 $T$.

Title | cyclic subspace |
---|---|

Canonical name | CyclicSubspace |

Date of creation | 2013-03-22 14:05:03 |

Last modified on | 2013-03-22 14:05:03 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 12 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 15A04 |

Classification | msc 47A16 |

Synonym | cyclic vector subspace |

Related topic | CyclicDecompositionTheorem |

Related topic | CyclicVectorTheorem |

Defines | cyclic vector |