# Group action on set SO(2)

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 $$SO(2)$$.

• Hi and welcome to MSE! Just a quick tip that will make it both easier for you to write and improve the formatting of your post: When using MathJax you do not need to use  around each individual math symbol. You can write an entire mathematical expression/equation within on pair of \$-s. May 1, 2021 at 8:22
• A group action is defined by specific properties that need to be satisfied. Did you try to check them? If so, where do you block? May 1, 2021 at 8:27

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.