Say I have a finite dimensional vector space $F^{d}$, and I have subspace of it, $V$, such that for each vector in it there is some coordinate which is zero, i.e. each vector is perpendicular to some basis vector.
What can be said about this space? Does it have some structure? Perhaps, all vectors are perpendicular to a common basis vector?
I tried to prove the last claim by Gauss-eliminating the basis matrix, but I didn't succeed. I did prove it for $n=2$.