Find linear dependence on these vectors

[ashina14]

Homework Statement

Suppose V = R^4 and let U = <X>, where X = {(1,0,-2,1),(2,-2,0,3),(0,2,-4,-1),(-1,2,-2,-2)}
Find linear dependence on X and use it to find a smaller generating set of U. Repeat the step until you reach a basis for U.

Homework Equations

The Attempt at a Solution

I have formed a matrix of the 4 vectors in X and reduced it to echelon form. I got
(1 | 0 | 2 | 1
0 | 1 | -1 | -1
0 | 0 | 0 | 0
0 | 0 | 0 | 0)

Let's say the 4 vectors were s, t, u, v respectively.
Then xs + xt + yu + zv = 0 is true for some constants w,x,y,z (1)

Then according to the REF form we can form two equations: w = 2y +z and x = -y+z
I thought I could substitute these into equation (1) and end up showing one vector as a combination of the others but I am not able to reach anywhere. Where am I wrong? Is there a better method to go about this?

 

[haruspex]

I would have thought that the relationship was implied by the steps to get the echelon form. I.e. in producing the null rows, you effectively executed the equation you're looking for.