Intersection of subspaces

Question

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.

Answer 1

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.