Reflection of Plane in Line through Origin is Linear Operator

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

Then the reflection $$s_M$$ of $$\mathbb{R}^2$$ in $$M$$ is the rotation of the plane in space through one half turn about $$M$$ as an axis.

$$s_M \circ s_M = I_{\mathbb{R}^2}$$ and hence $$s_M = s_M^{-1}$$.

If $$M$$ is the $x$-axis then $$s_M \left({\lambda_1, \lambda_2}\right) = \left({\lambda_1, -\lambda_2}\right)$$.

If $$M$$ is the $y$-axis then $$s_M \left({\lambda_1, \lambda_2}\right) = \left({-\lambda_1, \lambda_2}\right)$$.

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