Let S be the subset $$\{\left(1,0,2\right),\left(3,2,1\right),\left(1,-2,7\right)\}\subset\mathbb{R}^{3}$$ Find orthonormal bases for $S^{\perp}$ and $S^{\perp\perp}$
I have begun by putting these vectors in a matrix and reducing them to row echelon form which gives $$\begin{matrix} 1 & 0 & 2 \\ 0 & 1 & \frac{5}{2} \\ 0 & 0 & 0 \end{matrix}$$
So we see the maximimal linearly independent subset is $\{\left(1,0,2\right),\left(3,2,1\right)\}$ and thses vectors for a basis of span S. Where do I go from here?
Throught the Gram-Schmid pprocess (https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process) find an orthonormal basis of $S$ which will also be a basis for $S^{\perp\perp}$ (we can show this by the projection theorem also for infinite dimension vector spaces)
Of course, $\dim S^{\perp}=1$ so take a vector $z \in \mathbb{R}^{3}-S $ and take $\hat{z} $ its orthogonal projection on $S $
Then you have that $z-\hat{z} $ lives in $S^{\perp}$ and therefore is a basis