Let $\mathbb{R}$ be the field of real numbers and
$SO(2) := \{M \in SL_2(\mathbb{R})|\forall u,v \in \mathbb{R}^2: \langle Mu \rangle, \langle Mv \rangle = \langle u,v \rangle \}$,
where $\langle\ast,\ast\rangle$ denotes the standard scalar product on $\mathbb{R^2}$. a) Show that by
$\mu:SO(2) \times \mathbb{R^2} \rightarrow \mathbb{R^2}, (M,v) \mapsto Mv $
a group operation of $SO(2)$ on $\mathbb{R^2}$ is defined.
Can someone help me, please? I have no idea at all. I don't even understand the set $SO(2)$.