# Linear map problem

• February 10th 2013, 09:05 PM
jdm900712
Linear map problem
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

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.
• February 11th 2013, 07:08 PM
chiro
Re: Linear map problem
Hey jdm900712.

Can you make use of the rank-nullity theorem for your map?
• February 12th 2013, 01:07 AM
Deveno
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).