Let $T: V\to V$, and $V$ is of finite dimension.
$M$ is the minimal polynomial $M = M_1 \cdot M_2 \cdot\dots\cdot M_k$. $M_i$ monic polynomial. $V = W_1 \oplus W_2 \oplus \dots \oplus W_k$.
It's given that $W_i = Ker(M_i(T))$.
U is T invariant subspace. Prove that $U = (U\cap W_1)\oplus (U\cap W_2)\oplus\dots\oplus (U\cap W_k)$
It's clear to me that $W_i$ are invariant subspaces. I tried to use the fact that minimal polynomial of $T_u(=M_u)$ devides $M$.
But $M_u$ isn't necessarily $\prod M_i$. Can some one help me see what am I missing?