Reflection of Plane in Line through Origin is Linear Operator

Theorem
Let $M$ be a straight line in the plane passing through the origin.

Let $s_M$ be the reflection of $\R^2$ in $M$.

Then $s_M$ is the rotation of the plane in space through one half turn about $M$ as an axis.


 * $s_M \circ s_M = I_{\R^2}$

and hence:
 * $s_M = {s_M}^{-1}$

If $M$ is the $x$-axis then $\map {s_M} {\lambda_1, \lambda_2} = \tuple {\lambda_1, -\lambda_2}$.

If $M$ is the $y$-axis then $\map {s_M} {\lambda_1, \lambda_2} = \tuple {-\lambda_1, \lambda_2}$.

In general, $s_M$ is a linear operator for every straight line $M$ through the origin.