Hi Swati,Let T:V->V be a linear operator on an n-dimensional vector space. Prove that exactly one of the following statements holds:
(i) the equation T(x)=b has a solution for all vectors b in V.
(ii) Nullity of T>0
Yes of course. I should have said: Use the rank-nullity theorem again, to show that T is NOT surjective.if the SECOND statement is true, T CANNOT be surjective:
by the rank-nullity theorem:
dim(V) = rank(T) + nullity(T).
if nullity(T) > 0, then rank(T) < dim(V), so that:
dim(im(T)) < dim(V).
thus there is some b in V not in im(T).
(i only posted this because Opalg's post answers the wrong question).