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.