Continuous group action of $SO(2)$ on $\mathbb{R}^2$

Question

$SO(2)$ acts on $\mathbb{R}^2$ by complex multiplication, A continuous group action $\phi:SO(2)\times \mathbb{R}^2 \to \mathbb{R}^2$ defined by, $\phi(A,z)=Az=e^{i\theta}z=|z|e^{i(\theta+\theta_1)}$, where $arg(z)=\theta_1$ and $A=e^{i\theta}=\begin{bmatrix}\cos \theta & -\sin\theta \\ \sin \theta & \cos \theta \end{bmatrix}$

This is right? help me.

Answer 1

This is written in a very confused way. For instance, when you write$$e^{i\theta}=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix},$$you are stating that a complex number is equal to a $2\times2$ real matrix. Don't you see something strange in this statement?

The natural way of having $SO(2,\mathbb{R})$ acting on $\mathbb{R}^2$ is$$\left(\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix},(x,y)\right)\mapsto\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.(x,y).$$You don't need complex numbers to describe this action.