Linear Operator Proof (Show that $L(x)=ax$ for $Ɐxϵ\mathbb{R^1}$)

Question

I understand that $L$ maps from $\mathbb{R^1} \to \mathbb{R^1}$, but why does it sense to have $L(x1) = xL(1)$? Why would they be equivalent?

Answer 1

$L(xa)=xL(a)$ is part of the definition of a linear operator. (The other part is $L(a+b)=L(a)+L(b)$.)