Orthogonal basis of orthogonal complement Let $$U = \{\vec{x} \in \mathbb{R}^n : x_1=x_2=...=x_n\}$$
I have to find orthogonal basis of  $$U^{\bot}$$
I was trying to use the Gram–Schmidt process but I suppose it isn't working in this case. Is there any simple solution to this problem?

You want to find the vectors $\vec{y} = \left(\begin{array}{c} y_1 \\ \vdots \\ y_n\end{array}\right)$ such that $\left(\begin{array}{c} y_1 \\ \vdots \\ y_n\end{array}\right)\cdot\left(\begin{array}{c} 1 \\ \vdots \\ 1\end{array}\right) = 0$ (since every $\vec{x}\in U$ is a scalar multiple of this vector).
Thus what you have is the single equation $y_1 + \cdots + y_n = 0$. There are $n-1$ free parameters. Letting $y_n = -y_1 - \cdots - y_{n-1}$, can you see what a basis is?
