Let $S = \{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ be a set of vectors in $\mathbb{R}^{n}$. I would like to find a vector $\mathbf{u} \in \mathbb{R}^n$ such that, for all $i \in [1, n]$, $\mathbf{u}$ and $\mathbf{v}_i$ are not orthogonal. I was thinking about a linear combination of the vectors in $S$ but I'm not able to prove it.
Thanks!
EDIT: what if I assume also that the vectors in $S$ are also linearly independent? Would it make easier to find such a $\mathbf{u}$?
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$\begingroup$If $v_i$ is non-zero, then the set of vectors that are perpendicular to it consist of a $n-1$ hyper-plane.
Since we know that $n-$space cannot be covered by $n$ hyperplanes of dimension $n-1$, we know that (many) vectors exist.
In particular, if you take any set of points that cannot be covered by $n$ hyperplanes through the origin, then you're guaranteed that at least one of them will work.
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