hello :) I was trying to prove the following result : for a linear mapping L: V --> W dimension of a domain V = dimension of Im (L) + dimension of kernel (L) So, my doubt actually is that do we really need a separate basis for the kernel ? Theoretically, the kernel is a subspace of the domain V . So, the basis for V can be used to express any vector in the kernel . If we really need it, i think this might be the reason ( please check whether i am right or not ) The kernel directs the mapping to the zero vector. We are not interested in this trivial mapping. So, even though the basis vectors of V can clearly describe the vectors in kernel, we keep the vectors in kernel aside and are interested in only those vectors in V which do not produce the zero vector. Then, since T(ki) = 0 where ki represents any vector in the kernel , ki too ought to be a part of the solution for any mapping. Hence, a separate basis which only can express any vector in the kernel is needed Am i thinking on the right line ?