# Rotation of Plane about Origin is Linear Operator

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## 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:

\(\displaystyle r_\alpha \left({\lambda_1, \lambda_2}\right)\) | \(=\) | \(\displaystyle \tuple {\rho \cos \alpha \cos \sigma - \rho \sin \alpha \sin \sigma, \rho \sin \alpha \cos \sigma + \rho \cos \alpha \sin \sigma}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {\lambda_1 \cos \alpha - \lambda_2 \sin \alpha, \lambda_1 \sin \alpha + \lambda_2 \cos \alpha}\) |

The result follows from Linear Operator on the Plane.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 28$: Example $28.2$