Assume
show that:
If and
then
let and be the corresponding linear transformations defined by let's put and
since we have thus, using the rank nullity theore, we have and hence
on the other hand, since and we have, again by the rank-nullity theorem, now suppose
then and thus i.e. hence and therefore so
finally, using (1) and (2) and the rank-nullity theorem, completing the proof is easy: