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 $$r_\alpha: \mathbb{R}^2 \to \mathbb{R}^2$$ be the mapping defined as: $$\forall x \in \reals^2: r_\alpha \left({x}\right) =$$ the point into which a rotation of $$\alpha$$ 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.