Let V be a vector space over the field F. Let Hom(V,F) be the set of all homomorphisms of V into F (this is a pretty standard definition, noting new here.) Now, let f and g be functions in Hom(V,F). If f(v)=0 forces g(v)=0 then g = \lamda f for some \lamda in F.
No relevant equations
The Attempt at a Solution
So, since we are talking about mapping stuff to 0, it seems that we should look at the kernel of the transformations. We see that ker(f) is a subset (and hence sub-vector space) of ker(g). By homomorphism theorems, the quotient spaces formed by these kernels are both isomorphic to F, when F is being considered as a Vector Space. So V\ker(f) and V\ker(g) are both 1-D spaces and are isomorphic. So everything in each of the quotient spaces can be written as a multiple of a basis vector, v+ker(f) for V\ker(f) and u+ker(g) for v\ker(g) for some u,v in V. Ok, so since ker(f) in ker(g) it seems that everything in V\ker(f) can be written as a multiple of u+ker(g).
Other than that, I am kind of stuck on this. I need hints or ideas. I think perhaps I have built some sort of mental road block here. This problem is problem 4.4.11 from Topics In Algebra (Hernstein).