Group action on set SO(2)

Question

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 $SO(2)$.

Answer 1

Check the two properties of a group action:

1. $\mu(I, v)=Iv =v$ for all $v \in \mathbb{R}^2$.
2. $\mu(M, Nv)=M(Nv) = (MN)v = \mu(MN, v)$ for all $M, N \in SO(2)$ and all $v \in \mathbb{R}^2$.

Both of these are known facts from linear algebra about matrix-vector multiplication. I may also add that $SO(2)$ is not special (pun not intended) in this sense. The general linear group $GL(n, \mathbb{R})$ always acts on $\mathbb{R}^n$ by matrix-vector multiplication.