1. ## Linear map problem

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.

2. ## Re: Linear map problem

Hey jdm900712.

Can you make use of the rank-nullity theorem for your map?

3. ## Re: Linear map problem

what you've done is shown a) implies b).

now suppose we have b). so T2(v) = 0 implies v = 0.

suppose that w is in im(T) and ker(T). since w is in ker(T), T(w) = 0. since w is in im(T), w = T(v), for some vector v.

hence 0 = T(w) = T(T(v)) = T2(v).

by b), this means that v must be 0. hence w = T(v) = T(0) = 0, so every element of the intersection of ker(T) and im(T) is a 0-vector, that is: ker(T)∩im(T) = {0}, which is precisely a).