Dimensions: $\bigcap^{k}_{i=1}V_i \neq \{0\}$ Let $V$ be a vector space of dimension $n$ and let $V_1,V_2,\ldots,V_k \subset V$ be subspaces of 
$V$. Assume that
\begin{eqnarray}
\sum^{k}_{i=1} \dim(V_i) > n(k-1).
\end{eqnarray}
To show that $\bigcap^{k}_{i=1}V_i \neq \{0\}$, what must be done? Also, could there be an accompanying schematic/diagram to show the architecture of the spaces' form; that is, something like what's shown here. 

Hint: Can you do the case where $k = 2$? Then try induction on $k$, that involves replacing $V_1,V_2$ by $V_1 \cap V_2$, using the identity $\dim( V_1 + V_2) + \dim V_1 \cap V_2 = \dim V_1 + \dim V_2$.
