# Linear map problem

1. Feb 10, 2013

### jdm900712

1. The problem statement, all variables and given/known data
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) Im T $\cap$ Ker T = {0}
b) If T$^{2}$(v) = 0 -> T(v) = 0, v$\in$ V

2. Relevant equations

3. The attempt at a solution
Using p -> (q -> r) <-> (p$\wedge$q) ->r
I suppose Im T $\cap$ Ker T = {0} and T$^{2}$(v) = 0.
then I know that T(v)$\in$ Ker T and T(v)$\in$ Im T
so T(v) = 0.

I need help on how to prove the other direction.

2. Feb 10, 2013

### vela

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