### Instruction: Find the matrix that corresponds to rotating about the origin by an angle of $45^\circ$ clockwise.### Response: To rotate a point $(x,y)$ about the origin by an angle of $\theta$ clockwise, I can use the formula $(x \cos \theta + y \sin \theta, -x \sin \theta + y \cos \theta)$.
This means that the matrix that corresponds to this transformation is $\begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$.
Plugging in $\theta = 45^\circ$, I get $\begin{bmatrix} \cos 45^\circ & \sin 45^\circ \\ -\sin 45^\circ & \cos 45^\circ \end{bmatrix}$.
Using the fact that $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$, I can simplify this to $\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}$.