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.

A group representation is a group homomorphism $\phi:G\rightarrow \operatorname{GL}(V)$, where $V$ is a vector space.  If we take $V=\mathbb{R}^n$, we see that $\operatorname{GL}(V)=\operatorname{GL}_n\left(\mathbb{R}\right)$.  So, what you have got there is a real representation of finite dimension.
