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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.

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    $\begingroup$ the dimension of the intersection of subspaces is at most the smallest dimension of any of the subspaces involved. $\endgroup$ – Thomas Sep 28 '18 at 14:32
  • $\begingroup$ @Thomas sir, first of all thank you so much for your comment. Further is there is way of calculating dimension of the intersection of three subspaces. The above formula that I had mentioned can be used to calculate dimension of $W_1∩W2$ but how can we use it, to calculate dimension of intersection of three subspaces? $\endgroup$ – Akash Patalwanshi Sep 28 '18 at 14:38
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    $\begingroup$ math.stackexchange.com/questions/2065003/… Very closed to your question $\endgroup$ – StammeringMathematician Sep 28 '18 at 14:44
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$W$ is also a subspace of $W_i$, so $\dim W \leq \dim W_i = 4$.

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  • $\begingroup$ sir, thank you so much for your answer, Further is there is way of calculating dimension of the intersection of three subspaces. The above formula that I had mentioned can be used to calculate dimension of $W_1∩W2$ but how can we use it, to calculate dimension of intersection of three subspaces? $\endgroup$ – Akash Patalwanshi Sep 28 '18 at 14:41
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    $\begingroup$ See here or here or even here for some related discussion $\endgroup$ – Rodrigo Dias Sep 28 '18 at 17:17

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