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I am working on an example of vector spaces. I have the following question:

Let $\{V_1,V_2,\ldots,V_t\}$ be a family of $n$-dimensional subspaces and the dimension intersection of any $n$ subspace is at least one. Is it true that

$\dim \bigcap^{i=t}_{i=1}V_i\geq1$?

I have calculated $\dim \bigcap^{i=t}_{i=1}V_i$ for several spaces and all of spaces satisfied in $\dim \bigcap^{i=t}_{i=1}V_i\geq1$.

Can anybody take counterexample?

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    Each $V_i$ is $n$-dimensional. But you are taking exactly $n$ to take intersections?2012-06-30
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    @Sigur yes exactly2012-06-30

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