If $S=\{(1,0,i),(1,2,1)\}\subseteq X^3 $ then $S^{\bot}$ is span$\{(i,-\frac{1}{2}(1+i),1)\}$.

Question

If $S=\{(1,0,i),(1,2,1)\}\subseteq X^3 $ then $S^{\bot}$ is span$\{(i,-\frac{1}{2}(1+i),1)\}$.

This gives me 2 equations: $x+iz=0$ and $x+2y+z=0$ which gives me the general element of $S ^\bot$ as $z\left(-i,\frac{i-1}{2},1\right )$ but the given answer is span$\{(i,-\frac{1}{2}(1+i),1)\}$ and I'm not able to show that they are equivalent.

Answer 1

The standard dot product of two complex vectors $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is in fact not $x_1x_2+y_1y_2 + z_1z_2$, but rather $$ \overline{x_1}x_2+\overline{y_1}y_2 + \overline{z_1}z_2 $$ This is necessary to ensure that, for instance, the dot product of any vector with itself is a real (and non-negative) number.

Using this will give you slightly different equations, and presumably you get the given solution when you solve those.