There are two vector subspaces in $R^4$. $U1 := [(3, 2, 2, 1), (3, 3, 2, 1), (2, 1, 2 ,1)]$, $U2 := [(1, 0, 4, 0), (2, 3, 2, 3), (1, 2, 0, 2)]$

My idea was to calculate the Intersection of those two subspaces by putting all the given vector elements in a matrix (a vector is a column). If I run the gauss-algorithm, this leads to a matrix \begin{pmatrix} 1 & 0 & 0 & 0 & -6 & -4 \\\\ 0 & 1 & 0 & 0 & 3 & 2 \\\\ 0 & 0 & 1 & 0 &6 & 4 \\\\0 &0 & 0 & 1 & -1 & -1 \end{pmatrix}

So I see that the dimension of $U1 + U2$ equals 4, as there are 4 linear independent vectors. Is it somehow possible to get a basis of $[U1] \cap [U2]$ from this matrix? I know that it has to be one dimensional as the dimension of $U2$ equals 2.