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I am trying to prove the below but I am having problems starting and I think that I am misunderstanding something?

If we have that G is a connected linear group and $p:G\rightarrow GL(V)$ and ,V a finite dimensional vector space, is a homomrphism then I want to show that if $v\in V$ is fixed by every matrix in $G$ iff $Lp(X)v=0\ \forall X\in \mathfrak{g}$

So if $x\in\mathfrak{g}$ then we have that $\exp(tX)\in G$ by defintion and so:

$Lp(X)v=\frac{d}{dt}p(\exp(tX))v|_{t=0}$ now does this become $\frac{d}{dt}Xv|_{t=0}=0$?

Thanks for any help

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