Here is the question:
I am stuck on this question for more than a few hours. All my attempts to solve this question failed.
What I tried to do:
Show the following equality: $W = W \cap V = W \cap (W_1 \oplus W_2 ... \oplus W_k) = (W \cap W_1) \oplus ... \oplus (W \cap W_k)$
However, I couldn't manage to prove the last equality.
My next idea was to show that any $w \in W$ can be represented as a sum of k vector that belong to the intersections: $w = w_1 + w_2 ... + w_k$ such that $w_i \in (W \cap W_i)$ but I couldn't figure out how to do it for all possible T invariant subspaces.
Lastly, I would appreciate it if somebody could give me a hint or point me towards an idea. Thanks in advance :)
EDIT:
It seems a question very similar to this has already been asked here, and there is an existing answer in there.