Let $L$ be a simple $sl_2$-module of finite dimension over a field $K$ of characteristic $0$. Say $e,f,h$ is a basis of $sl_2$. In the notes I am reading it only defines the action of $e,f,h$ on the basis elements of $L$. From this I see that we get the action of $e,f,h$ on all elements of $L$.
I was wondering: Is the action of other elements of $sl_2$ determined by linearity? i.e. for example, given any $a, b \in K$ and $x \in L$, we define $$ (a e) \cdot (b x) := ab (e \cdot x)? $$
I was getting confused because it doesn't say anything about this. Thank you very much.