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:

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


The result follows from Linear Operator on the Plane.



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