Let V be an n-dimensional vector space where . If is a linear transformation, prove that the following statements are equivalent:

a) im(T)=ker(T)

b) , n is even and .

So far I have managed show show that (a) implies (b), but when trying to find (b) implies (a) I get stuck. This is my working so far:

dim(V) = r(T) + n(T)

so

, so T(V) ker(T)