theorem about cyclic subspaces

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

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