Exercise 4.1.9

• $f=1$4$

  $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right]$

  $, so we have

$x={\mathrm{\Omega }}_{4}f=\frac{1}{4}\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -i\hfill & \hfill -1\hfill & \hfill i\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill i\hfill & \hfill -1\hfill & \hfill -i\hfill \end{array}\right]\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \frac{1}{2}\hfill \\ \hfill 0\hfill \\ \hfill \frac{1}{2}\hfill \\ \hfill 0\hfill \end{array}\right]$

• $f=\frac{1}{4}\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]$, so we have

$y={\mathrm{\Omega }}_{4}f=\frac{1}{4}\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -i\hfill & \hfill -1\hfill & \hfill i\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill i\hfill & \hfill -1\hfill & \hfill -i\hfill \end{array}\right]\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]=\frac{1}{4}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill i\hfill \\ \hfill -1\hfill \\ \hfill -i\hfill \end{array}\right]$