Good job (Yuxin, Jilong). Jilong: please note that it is T0101, not T0201 (messing up tutorial sections not a big deal for us, but was a real monkey wrench thrown into in TT1 for MAT244)

$\renewcommand{\Re}{\operatorname{Re}}

\renewcommand{\Im}{\operatorname{Im}}$

Please escape sin, cos by \ : \sin, \cos (and so on) it will make them upright

Jingxuan is correct. Also if $f$ is real, then $\hat{f}$ is real iff $f$ is even, $\hat{f}$ is purely imaginary iff $f$ is odd.