Immediately, you know a=0, and that all 4-tuples that reside in U must then have zero first coordinate.

Take c and d free. Then b=2c-d. Now you can write, (a,b,c,d) = c(0,2,1,0) + d(0,-1,0,1).

So dim(U)=2, and a basis for U can be taken as {(0,2,1,0), (0,-1,0,1)}.

You can now express U by: <{(0,2,1,0), (0,-1,0,1)}>. which just says that U is the set of all linear combinations of the two guys inside the braces.