A function T:V→W between vector spaces V and W is called additive if T(x + y) = T(x) + T(y) for all x, y ∈V. Prove that if V and W are vector spaces over the field of rational numbers, then any additive function from V into W is a linear transformation.
My attempt to prove this would be to show that T(x) is closed under scalar multiplication because by definition then T would be linear, but I am not sure how to show this.