### 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 figure out how the standard basis vectors $\vec{i}$ and $\vec{j}$ are transformed by the rotation.
I can draw a picture of the coordinate plane and mark the points $(1,0)$ and $(0,1)$ that represent $\vec{i}$ and $\vec{j}$, respectively.
If I rotate the plane by $45^\circ$ clockwise, the point $(1,0)$ will end up at $(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$, and the point $(0,1)$ will end up at $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$.
These are the new coordinates of $\vec{i}$ and $\vec{j}$ after the rotation, so they form the columns of the matrix that I'm looking for.