# Intersection of subspaces

I am working on example of vector space. I have 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

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

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 any body take counter example? Thanks.

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 Each $V_i$ is $n$-dimensional. But you are taking exactly $n$ to take intersections? – Sigur Jun 30 '12 at 21:03 @Sigur yes exactly – Babgen Jun 30 '12 at 21:06 After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark ✓ next to it. This scores points for you and for the person who answered your question. If you don't do this, people are less likely to answer your later questions. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, How does accept rate work?. – Michael Albanese Dec 31 '12 at 15:08

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.