Rotation of Plane about Origin is Linear Operator

Theorem
Let $r_\alpha$ be the 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:
 * $\forall x \in \R^2: r_\alpha \left({x}\right) = \text { the point into which a rotation of } \alpha \text{ carries } x$

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

Proof
Let $\left({\lambda_1, \lambda_2}\right) = \left({\rho \cos \sigma, \rho \sin \sigma}\right)$.

Then:

The result follows from Linear Operator on the Plane.