Matrix for reflection about the line $y = \tan (\theta) \, x$ How would I show that a reflection about the line $y = \tan (\theta) \, x$ is the following?
\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}

The transformation\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}
maps any arbitrary vector $P=\begin{pmatrix}\alpha\\\beta\end{pmatrix}\in\mathbb R^2$ to the vector
$P'=\begin{pmatrix}\alpha\cos2\theta+\beta\sin2\theta\\\alpha\sin2\theta-\beta\cos2\theta\end{pmatrix}$.
It is easy to verify that the mid-point of line joining vectors $P$ and $P'$ i.e.
$M=\begin{pmatrix}\alpha\cos^2\theta+\beta\sin\theta\cos\theta\\\alpha\sin\theta\cos\theta+\beta\sin^2\theta\end{pmatrix}$
always lie on the line $y\cos\theta=x\sin\theta.$
