If the bottom holds in reverse, i.e. f(x) + f(y) = f(x+y) and af(x) = f(ax) does that imply that the map is linear, or does it only go in one direction?

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Is this Algebraical character unique in all tranformations?

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I usually found the geometrical way to think about problems in linear algebra and vector calculus is harder for me compared with the algebraical way. So is it necessary to think geometrically? If so, would you recommend some practices for it?

If the bottom holds in reverse, i.e. f(x) + f(y) = f(x+y) and af(x) = f(ax) does that imply that the map is linear, or does it only go in one direction?

Is this Algebraical character unique in all tranformations?

I usually found the geometrical way to think about problems in linear algebra and vector calculus is harder for me compared with the algebraical way. So is it necessary to think geometrically? If so, would you recommend some practices for it?