My main source of reference is https://en.wikipedia.org/wiki/Representation_of_a_Lie_group.
According to this link, if $G$ is a Lie group and $V$ a finite dimensional complex vector space, then we say that $(\pi,V)$ is a representation of $G$ if $\pi:G\to GL(V)$ is a smooth group homomorphism.
It is remarked that smoothness is a technicality, in that any continuous homomorphism will automatically be smooth.
The article then proceeds to say that can alternatively describe a representation of a Lie group $G$ as a linear action of $G$ on a vector space $V$. This is the part I am unsure of.
If we have such a linear action (say $\cdot)$, then I can see that $\pi:G\to GL(V)$ defined by $\pi(g)(v)=g\cdot v$ is a group homomorphism. However, why is smoothness satisfied? I suppose it suffices to show continuity, but this still does not seem clear to me.
Any help is appreciated!
Edit: In the case wikipedia is incorrect, would additional assumptions on $G$ (e.g. compactness) make the above definition correct?