I am trying to perform Haar wavelet transformation on the following vector which is defined over $GF(2)$.
$[1, 0, 1, 0, 1, 0, 1, 0]$
I am doing it as follows.
$[1, 0, 1, 0, 1, 0, 1, 0]$
$\implies [\frac{1 + 0}{\sqrt{2}}, \frac{1 + 0}{\sqrt{2}}, \frac{1 + 0}{\sqrt{2}}, \frac{1 + 0}{\sqrt{2}}, \frac{1 - 0}{\sqrt{2}}, \frac{1 - 0}{\sqrt{2}}, \frac{1 - 0}{\sqrt{2}}, \frac{1 - 0}{\sqrt{2}}]$
Now, we know that $\sqrt{2} \bmod 2 = 1 $.
$\implies [1 + 0, 1 + 0, 1 + 0, 1 + 0, 1 - 0, 1 - 0, 1 - 0, 1 - 0]$
$\implies [1 , 1, 1, 1 , 1 , 1, 1, 1]$
$\implies [\frac{1 + 1}{\sqrt{2}}, \frac{1 + 1}{\sqrt{2}}, \frac{1 - 1}{\sqrt{2}}, \frac{1 - 1}{\sqrt{2}} , 1 , 1, 1, 1]$
$\implies [0, 0, 0, 0, 1 , 1, 1, 1]$
$\implies [\frac{0 + 0}{\sqrt{2}}, \frac{0 - 0}{\sqrt{2}}, 0, 0, 1 , 1, 1, 1]$
$\implies [0, 0, 0, 0, 1 , 1, 1, 1]$
Did I do it write?