linear algebra equation set up

Here is my problem: Suppsoe that {u,v,w} is a linarly independent set in R^4

Is the set {u+v, 2u-v, w+v, u-w} linearly independent?

I did one of these that had a set in R^3 and for example it had {u,v,w} and the set was{u+v, u-v+2w, u+3v-2w} I used x, y,z

I stated that we know that if Au+bv+cw=0 then A=B=C=0

then x(u+v) + (x-y+3z)v + (2y_2z)w = o

my hint was A=u B=v C=w

and we know that:

A= x+y+z = 0 B= x-y+3z = 0 c= 2y -2z = 0

Then I set up my matrix and solved

I tried to follow the same logic and use x,y,z and s but when I got to the s( )

I noticed the u, v, w elements were all used up. Solving the matrix by row reduction was easy but as you know setting the problem up and going to the matrix and getting the answer from there does not help understand the part in the middle, which is where I am stuck. Any help would be appreciated.

Thank You very much!

Keith