### 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 rotating about the origin by an angle of $\theta$ clockwise is $\begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$.
To find the matrix for $\theta = 45^\circ$, I just need to plug in the values of $\cos 45^\circ$ and $\sin 45^\circ$.
I recall that $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$.
So the matrix for $\theta = 45^\circ$ is $\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}$.