a) Do this by contradiction. Suppose the result is false, so there is a nonzero vector z in the range of T (so z=Tx for some x in V), with Tz=0. From that, you should be able to show that the null space of T^2 is bigger than the null space of T. Then use the "rank + nullity" theorem to deduce that the range of T^2 is smaller than the range of T.

b) This is so badly formatted that I can't read the question. It looks as though it should be saying something like

Let be a linear transformation. Show that T is translation invariant if and only if for some .

Is that right? If so, what is ?