U $\subseteq$ $\mathbb R$$^5$ with span(v$_1$, v$_2$) and v$_1$= $\begin{pmatrix} 1 \\ 2 \\ 0 \\ 2 \\ 1 \\ \end{pmatrix}$ and v$_2$= $\begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ \end{pmatrix}$
Now I want to find an orthonormal basis for U$^{\bot}$.
The following is the "theoretical part", please verify that my thoughts on that are right.
Because $\mathbb R$$^5$ = U $\oplus$ U$^{\bot}$ the dimension of U$^{\bot}$ = 3.
I need the vectors b$_1$, b$_2$ and b$_3$ in the span of U$^{\bot}$ and all of them need to fulfill that $\langle$v$_1$,b$_i$$\rangle$ = $0$ and $\langle$v$_2$, b$_i$$\rangle$ = $0$ (and of course they are linear independent).
After I find my 3 vectors b$_i$ I need to set them all to $\Vert$b$_i$$\Vert$ = 1 with $\hat b_i$ = b$_i$ * 1/$\Vert$b$_i$$\Vert$. Now $\hat b_1$,$\hat b_2$ and $\hat b_3$ is the orthonormal basis of U$^{\bot}$.
My problem is I don't know how to work with this "theoretical part" to find those b$_1$, b$_2$ and b$_3$.
By inspection we can easily choose
such that $b_i\cdot v_j=0$ then we can orthonormalize them by GS process.
Otherwise we can use the vectors $e_i$ for the standard basis together with $v_1$ and $v_2$ as row vectors and obtain a basis by the RREF. Then we can orthonormalize that basis by GS process.