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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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    @Sigur yes exactly2012-06-30

1 Answers 1

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Consider the vector space $\mathbb{R}^3$. Let $t = 3$ and $n = 2$. Consider $V_1$ to be the $xy$ plane. $V_2$ to be the $xz$ plane. $V_3$ to be the $yz$ plane. All of these are $2$ dimensional subspaces. The intersection of any pair is one dimensional line. However the intersection of all three is just the origin which is not $1$-dimensional.

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    ok thanks,it is ve$r$y helpful2012-06-30