Question: let $W_1, W_2, W_3$ be three distinct subspaces of vector space $\mathbb{R^{12}}$ each having dimension $4$, and let $W=W_1 ∩W_2 ∩ W_3$ then which of the following is/ are correct?
(1) $W$ is subspace of $\mathbb{R^{12}}$
(2) $dim(W)< 8$
(3) $dim(W)>7$
(4) $dim(w)<10$
My attempt: clearly $W$ will be subspace of $V$ (as intersection of subspaces is again subspace) hence (1) is true.
I stuck on other options, but I know $dim(W)≤dimV=12$. Further, I know if $W_1, W_2$ are subspaces of finite dimensional vector space then,
$dim(W_1+W_2)= dim(W_1)+dim(W_2)-dim(W_1∩W_2)$
But, how to use this formula, as we had intersection of three subspaces? further, using above formula is useful here? Please help needed.
Best Answer
$W$ is also a subspace of $W_i$, so $\dim W \leq \dim W_i = 4$.