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: \map {r_\alpha} x = \text { the point into which a rotation of $\alpha$ carries $x$}$

Then $r_\alpha$ is a linear operator.

Proof

Let $P = \tuple {\lambda_1, \lambda_2}$ be an arbitrary point in $\R^2$.

From Equations defining Plane Rotation:

$\map {r_\alpha} P = \tuple {\lambda_1 \cos \alpha - \lambda_2 \sin \alpha, \lambda_1 \sin \alpha + \lambda_2 \cos \alpha}$

This demonstrates that $r_\alpha$ can be expressed as an ordered tuple of $4$ real numbers.

The result follows from Linear Operator on the Plane.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.2$