Question 3683
 if a(u + v + w)+ b(v + w) + c w = 0 for scalars a,b,c
 then a u + (a+b)v + (a+b+c)w = 0
 since u,v,w are independent 
 we have a = 0, a+b = 0 and a+b+c = 0

 This implies b = 0 and so c = 0.
 This shows u + v + w, v + w, w are three independent
 and so {u + v + w, v + w, w} forms a basis for V,
 because {u,v,w} is a basis of V, dim V = 3.

 Kenny