Rotation of Plane about Origin is Linear Operator

Theorem
Let $r_\alpha$ be the plane rotation of the plane about the origin through an angle of $\alpha$.

That is, let $r_\alpha: \R^2 \to \R^2$ be the mapping defined as:
 * $\forall x \in \R^2: \map {r_\alpha} x = \text { the point into which a rotation of } \alpha \text{ carries } x$

Then $r_\alpha$ is a linear operator determined by the ordered sequence:
 * $\tuple {\cos \alpha -\sin \alpha, \sin \alpha + \cos \alpha}$

Proof
Let $\tuple {\lambda_1, \lambda_2} = \tuple {\rho \cos \sigma, \rho \sin \sigma}$.

Then:

The result follows from Linear Operator on the Plane.