Q: True or False $W_i\cap \Sigma_{i\neq j}W_j=\Sigma_{i\neq j}( W_i\cap W_j)=\cap W_i$ Is the following true?
For vector spaces $W_1,...,W_m$
$$ W_i\cap \Sigma_{i\neq j}W_j=\Sigma_{i\neq j}( W_i\cap W_j)=\cap W_i $$
Thank you.
Edit:
I'm trying to prove that if $V=\Sigma_{i}W_i$ and zero  vector of $V$ has a unique representation of vectors of  $W_i$, then  $W_i\cap \Sigma_{i\neq j}W_j=\{0\}$

A: Try in $\mathbb{R}^2$ $W_1 = < (1,0) >, W_2 = < (0, 1) >, W_3 = < (1,1)>$. Then $W_2 + W_3 = \mathbb{R}^2$, and so $$W_1 \cap (W_2 + W_3) = W_1$$but $$\cap W_i = 0$$
