### Instruction: Find the matrix that corresponds to rotating about the origin by an angle of $45^\circ$ clockwise.### Response: To find the matrix that corresponds to a rotation, I need to think about how the standard basis vectors $\vec{i}$ and $\vec{j}$ are transformed by the rotation.
The vector $\vec{i} = (1, 0)$ is rotated $45^\circ$ clockwise to become $\vec{i}' = (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$.
Similarly, the vector $\vec{j} = (0, 1)$ is rotated $45^\circ$ clockwise to become $\vec{j}' = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.
Now I can use these new vectors as the columns of the matrix that represents the rotation.