Matrix Representation of Linear Rotation

Recall that Linear Transformation Rotation is: $R_{\theta }(x,y)=(x\cos (\theta )-y\sin (\theta ),y\cos (\theta )+x\sin (\theta ))$ We calculate $R_e$ on the standard basis: $R_{\theta }=(1,0)=(\cos (\theta ),\sin (\theta ))$ $R_{\theta }(e_1)=\cos (\theta )e_1+\sin (\theta )e_2$ $R_{\theta }(0_{11})=(-\sin \theta ,\cos \theta )$ $R_{\theta }(e_2)=(-\sin \theta )e_1+\cos (\theta )e_2$ This gives:

$$[R_{\theta }{]}_{\alpha }^{\alpha }=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$$