Reflection of Plane in Line through Origin is Linear Operator

Theorem
Let $M$ be a straight line in the plane $\R^2$ passing through the origin.

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

Then $s_M$ is a linear operator for every straight line $M$ through the origin.

Proof
Let the angle between $M$ and the $x$-axis be $\alpha$.

To prove that $s_M$ is a linear operator it is sufficient to demonstrate that:
 * $(1): \quad \forall P_1, P_2 \in \R^2: \map {s_M} {P_1 + P_2} = \map {s_M} {P_1} + \map {s_M} {P_2}$
 * $(2): \quad \forall \lambda \in \R: \map {s_M} {\lambda P_1} = \lambda \map {s_M} {P_1}$

So, let $P_1 = \tuple {x_1, y_1}$ and $P_2 = \tuple {x_2, y_2}$ be arbitrary points in the plane.

and:

Hence the result.