### 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 $(x',y') = (x \cos \theta + y \sin \theta, -x \sin \theta + y \cos \theta)$.
This means that if I have a vector $\begin{pmatrix} x \\ y \end{pmatrix}$, I can multiply it by the matrix $\begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$ to get the rotated vector $\begin{pmatrix} x' \\ y' \end{pmatrix}$.
In this case, $\theta = 45^\circ$, so I need to plug in $\cos 45^\circ = \frac{\sqrt{2}}{2}$ and $\sin 45^\circ = \frac{\sqrt{2}}{2}$ into the matrix.
This gives me the matrix $\begin{pmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix}$.