Let V be a vector space over the field F. and T L(V, V) be a linear map.

Show that the following are equivalent:

a) Im T Ker T = {0}

b) If T^2(v) = 0 -> T(v) = 0, v V

Using p -> (q -> r) <-> (p q) ->r

I suppose Im T Ker T = {0} and T (v) = 0.

then I know that T(v) Ker T and T(v) Im T

so T(v) = 0.

I need help on how to prove the other direction.