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.

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    $\begingroup$ preserving the vector space structure means that your function is linear $\endgroup$ – Dominic Michaelis Sep 30 '15 at 12:58
  • $\begingroup$ A homomorphism of vector spaces over a field $F$ is an $F$-linear map by definition. There is nothing to prove. $\endgroup$ – anon Sep 30 '15 at 12:59
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    $\begingroup$ Why don't you write down the definition of vector space homomorphism and linear transformation? If you write it down, you will automatically find your answer. $\endgroup$ – Babai Sep 30 '15 at 13:13
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    $\begingroup$ My question is really more like why we define a VS homomorphism to be a linear transformation, given the more general notion of a homomorphism. I think I've got it figured out though. A function which preserves the VS structure should preserve the two operations which define a VS. That is, for some homomorphism $f$, we should have $f(av) = af(v)$ and $f(v+w)=f(v)+f(w)$. But that exactly means that $f$ is linear. $\endgroup$ – user275324 Sep 30 '15 at 13:16
  • $\begingroup$ @user275324 That is absolutely correct. $\endgroup$ – Omnomnomnom Sep 30 '15 at 16:58

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