# theorem about cyclic subspaces

Let $k$ be field, $V$ a vector space^{} over $k$, $dimV=n$, and $T:V\to V$ a linear operator^{}. Let $W$ be a subspace^{} of $V$. And let ${v}_{1},\mathrm{\dots},{v}_{r}\in V$ such that $W=Z({v}_{1},T)\oplus \mathrm{\cdots}\oplus Z({v}_{r},T)$ (see the cyclic subspace definition), and $({m}_{{v}_{i}},{m}_{{v}_{j}})=1$ if $i\ne j$, where ${m}_{v}$ denotes the minimal polynomial of $v$ (or in other words, its annihilator polynomial). Then, $Z({v}_{1}+\mathrm{\cdots}+{v}_{r},T)=Z({v}_{1},T)\oplus \mathrm{\cdots}\oplus Z({v}_{r},T)$, and ${m}_{{v}_{1}+\mathrm{\cdots}+{v}_{r}}={m}_{{v}_{1}}\mathrm{\cdots}{m}_{{v}_{r}}$.

Title | theorem about cyclic subspaces |
---|---|

Canonical name | TheoremAboutCyclicSubspaces |

Date of creation | 2013-03-22 14:15:16 |

Last modified on | 2013-03-22 14:15:16 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 12 |

Author | Mathprof (13753) |

Entry type | Theorem |

Classification | msc 15A04 |