So we have transformation T:V->W and I proved that dimV=dimkerT+dimImT and there is a claim saying that if dimKerT=0 then our transformation is one to one function, and it's easy to prove however what I don't understand is , there are times that this doesnt work, What are the conditions for that? V has to be a finite vector space, in other word, that can be represented by finite number of liniearly independent vectors? or does the domain and the codomain of the function has to be the same vector space? Or it doesn't matter what the vector spaces are?