I'll try to write this as best as I can...
Let the following $U_1, U_2$ be subspaces of $\mathbb{R}^4$
$ U_1 = \begin{Bmatrix} (x, y, z, w) : z-y+2w = 0 \end{Bmatrix} $
$ U_2 = \begin{Bmatrix} (x, y, z, w) : z-y+2w = 0, x=2z \end{Bmatrix} $
Find a basis for the subspace $(U_1 \cap U_2)$
I have found the bases
$ B_1 = \begin{Bmatrix} (1, 1, 0, 0), (0, 2, 0, 1), (0, 0, 1, 0) \end{Bmatrix} $ $ B_2 = \begin{Bmatrix} (2, 2, 1, 0), (0, 2, 0, 1) \end{Bmatrix} $
for $U_1, U_2$ respectively, but do not know where to go from here, any help would be greatly appreciated.