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

Show that the following are equivalent:

a) Im T $\displaystyle \cap$ Ker T = {0}

b) If T^2(v) = 0 -> T(v) = 0, v$\displaystyle \in$ V

Using p -> (q -> r) <-> (p$\displaystyle \wedge$q) ->r

I suppose Im T $\displaystyle \cap$ Ker T = {0} and T$\displaystyle ^{2}$(v) = 0.

then I know that T(v)$\displaystyle \in$ Ker T and T(v)$\displaystyle \in$ Im T

so T(v) = 0.

I need help on how to prove the other direction.