Yes, for part A I meant that in order to find the basis for the kernel I would need to do
T(u-v) = T(U) - T(V). The vectors that would go to [0 0 0 0] under the transformation are
[0,1,-1,0] and [1,0,0,-1].
So the basis for the kernel of this transformation is B= {[0,1,-1,0], [1,0,0,-1]}
How...