Hi, if anyone could either confirm whether my answers to these questions are right, or if not then help me find the correct answer, I'd be very grateful.

U = {(0,0,a,b) : a,b in R }

V = {(x,y,z,w) : x=2z, x+y+w=0 }

W = {(x,y,z,w) : x+y+w=0 }

Question (a): Find bases for U, V, W.

I found bases of U to be (0,0,1,0) and (0,0,0,1) - which, I hope, is simple enough.

For V I rearranged to { a(2,-1,1,-1) : a in R } following the conditions set, so does this make (2,-1,1,-1) a basis?

Similarly for W, i rearranged to { a(2,-1,0,-1) + b(0,0,1,0) : a,b in R} since z isn't specified as relating to x, y, or w in the conditions as in V. So this makes (2,-1,0,-1) and (0,0,1,0) a basis?

Question (b): Which of the sums U+V, U+W, V+W are direct?

For the sums to be direct I know that the union of the two subspaces must be {0}, but how do I prove that? Am I right in thinking that since V is a subspace of W the sum is not direct?

Question (c): Which of the sums above equal R4?

Is this when the dimension of the sums equals 4? So I have dim(U)=2, dim(V)=1 and dim(W)=2.

Using the formula dim(U+W) = dim(U) + dim(W) - dim(UnW) , so if the sum is direct I can ignore the dim(UnW) part?

Any help would be much appreciated, thanks.