f(r,s,t)=(-r+3s+2t) is a linear map (from R^3 to R), so its kernel (U) is a subspace of the domain.
Or just show that the set is closed under addition and scalar multiplication.
If [r, s, t] is in that set then -r+3s+2t=0. If [u, v, w] is in that set, then -r+ 3v+ 2w= 0.
What about [r+ u, s+ v, t+ w]? Does it satisfy that equation?
What about [ar, as, at] for some number a? Does it satisfy that equation?