>[!Note] Definition (Rotation Matrix for $\mathbb{R}^{2}$) >The [[Matrix|matrix]] $A= \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$is a rotation matrix. # Properties >[!Note] Theorem ($A$ rotates any non-zero vector by $\theta$) >Note that the corresponding [[Left Multiplication Linear Map of Real Matrix|left multiplication linear map]] $L_{A}:\mathbb{R}^{2} \to \mathbb{R}^{2}$ takes the standard bases to $L_{A}(\underline{e}_{1}) = A \underline{e}_{1} = \begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix} \quad \text{and} \quad L_{A}(\underline{e}_{2}) = \begin{pmatrix} -\sin \theta \\ \cos \theta \end{pmatrix}$ We see the standard basis rotate by $\theta$ anticlockwise and indeed every other vector does the same as we naturally imagine. >*Proof*. For all $\underline{v} \in \mathbb{R}^{2} \setminus \{ \underline{0} \}$, we can check the [[Euclidean Norm|length]] of its image ${\underline{w}} = L_{A} (\underline{v}) = (x\cos \theta + y\sin \theta , -x \sin \theta + y \cos \theta)^{T}$ is unchanged: $||L_{A}(\underline{v}) || = \sqrt{ x^{2}(\sin^{2} \theta +\cos ^{2} \theta) +y^{2}( \sin^{2} \theta + \cos^{2} \theta) } = \sqrt{ x^{2}+y^{2} } = || \underline{v} ||$ >Also the [[Angle Between Nonzero Real Vectors|angle]] between them is $\angle v \,A \underline{v} = \cos^{-1} (\underline{\hat{v}} \cdot \underline{\hat{w}}) = \cos^{-1} \left( \frac{x^{2}\cos \theta + y^{2} \cos \theta}{x^{2} +y^{2}} \right) = \theta $So $L_{A}$ is an anticlockwise rotation about $\underline{0}$ by $\theta$. >*Proof*. Follows from [[Linear maps#^a8ff22|Linear Maps are equal if they agree on basis elements]]. > [!NOTE] Lemma ($A$ is invertible) > Note that $A$ is [[Inverse of Real Square Matrix|invertible]]. See Also [[Plane Reflection Matrix]].