I am dealing with representations of Lie groups, and as one knows, by the representation of a Lie group G, we refer to the following:
$1)$ (geometrical apect)
A representation of the Lie group $G$ on the (finite-dimensional complex) vector space $V$ is a continuous action $\phi: G$x$V \to V$ of $G$ on $V$such that for each $g \in G$ the translation $I_g: v \to \phi(g,v)$ is a linear map.
$2)$ (numerical aspect)
A matrix repesentation of $G$ is, simply, any continuous homomorphism $I: G \to GL(n,\mathbb C)$.
Understanding these two approaches, can someone give me two examples for these two definitions; one for each of them.
Thanks in advance