Show that every element on $O(\mathbb{R} ^2)$ is either a rotation or reflection

Question

Where $O(\mathbb{R} ^2)$ is the orthogonal group of $\mathbb{R} ^2$ or;

The set of all linear maps $g: \mathbb{R} ^2 \rightarrow \mathbb{R} ^2$ represented by an $n \times n$ matrix $M$ w.r.t. the standard basis of $\mathbb{R} ^2$ such that $M M^T$ is the identity matrix

I'm really not sure how to go about this?

Answer 1

Here is an idea : You need to first show that the orthogonal matrices in $\mathbb R^2$ are of the form either

\begin{bmatrix} \cos \theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{bmatrix} or

\begin{bmatrix} \cos \theta & \sin\theta \\ \sin\theta & -\cos\theta \\ \end{bmatrix} Hence you can conclude that either it is a reflection or a rotation . I hope that helps .