The matrix $A =\begin{pmatrix} \cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \end{pmatrix}$is the reflection in a line at angle $\theta /2$ above the $x$-axis. >*Proof*. >*Proof*. Simply check that $A$ reflects standard basis elements since [[Linear maps#^a8ff22|Linear Maps are equal if they agree on basis elements]]. >*Proof*. You can also check that $A^{2} = I_{2}$, so that $L_{A}$ is an *involution* of the plane (which, by definition, simply means that if you do it twice you get back to where you started). > [!NOTE] Lemma ($A$ is invertible) > Note that $A$ is [[Inverse of Real Square Matrix|invertible]]. See Also [[Plane Rotation Matrix]].