I'm doing this problem, which gives me these subspaces of $\mathbb{R}^4$
$$U=\text{span}\left\{\;\begin{pmatrix} 3\\ 2\\4 \\ -1\end{pmatrix},\;\begin{pmatrix} 1\\ 2\\1 \\ -2\end{pmatrix},\;\begin{pmatrix} -2\\ 3\\3 \\ 2\end{pmatrix}\;\right\}\qquad V=\text{span}\left\{\;\begin{pmatrix} 1\\ 4\\2 \\ 4\end{pmatrix},\;\begin{pmatrix} -1\\ 1\\-1 \\ -1\end{pmatrix},\;\begin{pmatrix} 3\\ 1\\2 \\ 0\end{pmatrix}\;\right\}$$
I'm meant to find a basis for $U\bigcap V$. From the dimension of subspace equation I know that it has dimension $2$. 

I first tried playing around with the bases for $U$ to get something in $V$, and vice versa, but I just can't spot anything... The only other option I can think of is to write down a system of equations, but this would have $6$ unknowns and $4$ equations which doesn't feel like a good approach.

Is there another way of doing this? Or have I just missed a fairly obvious trick?