A question about Lie group homomorphisms Suppose I have a Lie group $G$ and a Lie homomorphism $ \phi : G \rightarrow GL_n(\mathbb{R})$.
Can $ \phi $ be viewed as some sort of representation of $G$?
Can anyone make this rigorous for me please?
Thanks.

Another common way of defining a representation is that it's a vector space $V$ together with a smooth map $G \times V \to V$ that satisfies the axioms of an action by linear transformations.
First convince yourself that the canonical action $\operatorname{GL}_n(\mathbb R) \times \mathbb R^n \to \mathbb R^n$ is smooth.  Then given $\phi$ as above you have the map
$$G \times \mathbb R^n \xrightarrow{\phi\times\operatorname{id}}\operatorname{GL}_n(\mathbb R) \times \mathbb R^n \to \mathbb R^n$$
which is a composition of smooth maps, so it's smooth and gives the action that you're looking for.
