Let V be an n-dimensional vector space where . If is a linear transformation, prove that the following statements are equivalent:
a) im(T)=ker(T)
b) , n is even and .
So far I have managed show show that (a) implies (b), but when trying to find (b) implies (a) I get stuck. This is my working so far:
dim(V) = r(T) + n(T)
so
, so T(V) ker(T)