Show that $ (\pi(g)\phi)(v)=\phi({^t}gv) $ defines a representation

Question

Let $ G=SL_2(\mathbb{C}) $ and consider the action of $ G $ on the space of smooth functions on column vectors $ v \in \mathbb{C^2} $ given by: $ (\pi(g)\phi)(v)=\phi({^t}gv) $

Question 1: Show that $ \pi $ defines a representation

I must show that $\pi(gh)=\pi(g)\pi(h)$ for $g, h \in G$

I have had a go at playing around with the formula but am not sure how to manipulate $ (\pi(g)\phi)(v)=\phi({^t}gv) $

Could you give me some pointers please?

Answer 1

$$(g.(h.\phi))(v)=(h.\phi)(^tg.v)=\phi(^th.(^tg.v))=\phi((^th^tg).v)=\phi(^t(gh).v)=((gh).\phi)(v).$$