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 $SO(2)$.

  • $\begingroup$ 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. $\endgroup$ May 1, 2021 at 8:22
  • $\begingroup$ 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? $\endgroup$
    – Taladris
    May 1, 2021 at 8:27

1 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.


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