Exercise 6.2.9

The orthonormal basis for $W$ is the set consisting of the normalized vector $(i,0,1)$, $\{\frac{1}{\sqrt{2}}(i,0,1)\}$. To find a basis for $W^{\text{⊥}}$ is to find a basis for the null space of the following system of equations

The basis would be $\{(1,0,i),(0,1,0)\}$. It’s lucky that it’s orthogonal. If it’s not, we should apply the Gram-Schmidt process to it. Now we get the orthonormal basis

by normalizing those elements in it.