linear algebra problem: consequences of im(T)=ker(T) ?

I can't work out how to do this proof.

V is a vector space of dimension n where n >/= 1.

T: V --> V is a linear transformation.

You need to show that the following two are equivalent:

(a) im(T)=ker(T)

(b) T^2 = 0 AND n is even AND rk(T)=n/2

I think once you have proved that (a) implies (b), the other way round would be easy, and i've already shown that (a) implies T^2=0, but could anyone help me on how to show the other two parts of (b)?

Thanks very much