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Rotation
Example. Consider a linear function with
- $A(e_1) = e_2$;
- $A(e_2) = -e_1$.
Then $$A(\alpha e_1 + \alpha_2e_2) = \alpha_1A(e_1) + \alpha_2A(e_2)$$ $$ = \alpha_1e_2 - \alpha_2e_1$$ $$ = -\alpha_2e_1 + \alpha_1e_2.$$ This is an example of rotation!
Rotation is a linear transformation of the plane.
Example. Rotation, rotated through $\alpha$:
\[ R = \left| \begin{array}{ccc}
\lambda \cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha \end{array} \right|.\]
Then $$\det R = \cos ^2 \alpha + \sin ^2 \alpha = 1,$$ no stretching.
Exercise. What is $R$ in polar coordinates?
Theorem. All rotations are homotopic.