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 $m_{v_{1}+\cdots+v_{r}}=m_{v_{1}}\cdots m_{v_{r}}$.