R^2/Rotation/Definition

Rotation

A linear mapping

$D(\alpha )\colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2},$

which is given by a rotation matrix ${}{\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}$ (with some ${}\alpha \in \mathbb {R}$) is called rotation.