Suppose $V=V_1\oplus ...\oplus V_n $. Let $T\in$ Hom$(V,V)$ such that $T(V_i)\subseteq V_i$ for all $i=1,...,n$. Find a basis $\alpha$ of $V$...

Question

I know that a basis for $V$ is of the form $B=B_1\oplus... \oplus B_n$ where every $B_i$ is base for $V_i$. By hypothesis, I have to $M_i(V_i)\subseteq V_i$ for every $i=1,...,n$. But I could not conclude from this that $B_i=M_i$? what you think?