I think to show two statements are equivalent, it suffices to show that a imples b and b imples a. Pf: First part Assume Intersection of Range(T) and Ker(T)[ which is the null space] ={0} WTS: if T(T(x))=0, then T(x)=0 [using x instead of alpha] now, assume T(T(x))=0. This implies T(x) is an element of Ker(T) by the definition of Kernel. But T(x) is also an element of Range of T by the definition of Range. This implies T(x) is in the intersection of R(T) and Ker(T). However, by our initial assumption, the intersection has only one element which is 0. Therefore T(x) must be 0. Second Part Other way round is similar, Assume if T(T(x))=0, then T(x)=0 WTS: Intersection of Range(T) and Ker(T) ={0} By definition of Range T(x)=0 is an element of Range(T). Since T(T(x))=0, T(x)=0 is in the Ker(T). Since T(x)=0 is in both the Range and Ker, it is in the intersection of R(T) and Ker(T).