I understand how you did this problem, but I am confused about your anser to part b:
Shouldn't this be (a,a,0,-a)+(0,-b,b,b)+(c,0,c,0)=a(1,1,0,-1)+b(0,-1,1,1)+c(1,0,1,0)
Set={(1,1,0,-1),(0,-1,1,1),(1,0,1,0)} for all vectors of the form (a+b, a-b, b+c, -a+b)? Thank you!