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## Main Question or Discussion Point

The definition of a homomorphism is that it must preserve some algebraic structure, so if I transform a vector space using homomorphism between vector spaces (linear map), the result must be a vector space too, correct?

Now, if "v" and "w" are two vectors in a vector space V, than "(v + w)" must be a vector in V as well. And if "f(v)" and "f(w)" are vectors in a vector space V', than "f(v) + f(w)" must also be in V'.

But the definition of a linear map is that "f(v + w) = f(v) + f(w)". Would't it suffice to say that if "f(v)" and "f(w)" are in the resulting vector space, "f(v + w)" must be as well? Why must "f(v + w)" equal "f(v) + f(w)"?

If my question seems stupid in the context of linear algebra just think of it in terms of groups and homomorphisms between groups. That's where my confusion started anyway.

Now, if "v" and "w" are two vectors in a vector space V, than "(v + w)" must be a vector in V as well. And if "f(v)" and "f(w)" are vectors in a vector space V', than "f(v) + f(w)" must also be in V'.

But the definition of a linear map is that "f(v + w) = f(v) + f(w)". Would't it suffice to say that if "f(v)" and "f(w)" are in the resulting vector space, "f(v + w)" must be as well? Why must "f(v + w)" equal "f(v) + f(w)"?

If my question seems stupid in the context of linear algebra just think of it in terms of groups and homomorphisms between groups. That's where my confusion started anyway.