Find an orthonormal basis $u_1, u_2, u_3$ for $\mathbb{C}^3$ such that $u_1$ is a multiple of $(1, w, w^2)$, where $w=e^\frac{2i\pi}{3}$

Question

"Find an orthonormal basis $u_1, u_2, u_3$ for $\mathbb{C}^3$ such that $u_1$ is a multiple of $(1, \omega, \omega^2)$, where $\omega=e^\frac{2i\pi}{3}$"

An example that I came across that I cannot figure out, and the solution doesn't make much sense to me either.

I understand that I need to use Gram-Schmidt here, and I can usually do this given standard vectors, however I am unsure how to go about this.

Thank you

Answer 1

Let $v_1=(1,\omega,\omega^2)$ and pick $v_2$ and $v_3$ such that $\{v_1,v_2,v_3\}$ is a basis of $\mathbb{C}^3$. Better still, choose them so that, besides $\{v_1,v_2,v_3\}$ being a basis of $\mathbb{C}^3$, they are orthogonal to $v_1$ (with respect to the inner product defined by $\bigl\langle (z_1,z_2,z_3),(w_1,w_2,w_3)\bigr\rangle=z_1\overline{w_1}+z_2\overline{w_2}+z_3\overline{w_3}$). For instance, take $v_2=(-\overline\omega,1,0)$ and $v_3=(-\overline{\omega^2},0,1)$. Then apply Gram-Schmidt to $\{v_1,v_2,v_3\}$.