Linear Map Problem: Proving a and b Equivalent

[jdm900712]

Homework Statement

Let V be a vector space over the field F. and T $\in$ L(V, V) be a linear map.
Show that the following are equivalent:

a) I am T $\cap$ Ker T = {0}
b) If T$^{2}$(v) = 0 -> T(v) = 0, v$\in$ V

Homework Equations

The Attempt at a Solution

Using p -> (q -> r) <-> (p$\wedge$q) ->r
I suppose I am T $\cap$ Ker T = {0} and T$^{2}$(v) = 0.
then I know that T(v)$\in$ Ker T and T(v)$\in$ I am T
so T(v) = 0.

I need help on how to prove the other direction.

 

[vela]

Can you prove {0} ⊂ I am T ∩ Ker T? If so, all you have left to show is I am T ∩ Ker T ⊂ {0}.