Finding a basis for the intersection of two subspaces

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.

Solution 1:

Since $U_{2}\subset U_{1}$, you have $U_{1}\cap U_{2}=U_{2}$ and for $U_{2}$ for you already found a base (incase that's calculated correctly). I got a different base for $U_{2}$ but there are plenty of different bases for it.