How to show if $F(\alpha u + v) = \alpha F(u) + F(v)$ then $F$ is a linear map?

Question

Consider the function $F : V \rightarrow W$.

Assume $F(\alpha u + v) = \alpha F(u) + F(v)$ holds true for any $u,v \in V$ and scalar $\alpha$. Show that $F$ is linear, that is

$$ F(a+b)= F(a) + F(b) \,\,\,\, \forall a, b \in V $$ and $$ F(ca)= cF(a) \,\,\,\, \forall a \in V , c \in \mathbb{R} $$

Hint: Show $F(0) = 0$

Answer 1

Hint Setting $u=v=0$ you get $F(0)=0$.

Next, setting $u=a, v=b, \alpha =1$ you get the first relation.

Setting $v=0$ you get the second relation.