What is the basis we have for saying that the set of homomorphisms from vector spaces $V$ to $W$ is the set of linear transformations from $V$ to $W$? Is there a standard proof to show that the set of functions which map $V$ to $W$ such that $W$ is still a vector space (i.e. such that the vector space "structure" is preserved) is exactly the of linear functions from $V$ to $W$?
I might be able to prove this myself if I could figure out exactly what "preserving the VS structure" means. As it stands, it is too vague to me for me to see how to test for it.