1. The problem statement, all variables and given/known data Prove for every subspace B of vector space C, there is at least 1 linear operator L: C→C with ker (L) = B and there's at least 1 linear operator L':C→C with L'(C) = B. 2. Relevant equations 3. The attempt at a solution The first operator with Ker(L) = B would be anything mapping something to 0. So this would map to 0. So by the definition of a vector space, there exists a 0 element. And since the definition of a subspace is something that has the same conditions as the vector space. Therefore, the kernel exists in both? And for the second operator, this would be the identity element. So would the same argument as above work?