= Maximum-support vector in a finite-dimensional subspace
For any <field> $F$ and <vector subspace> $W\subseteq F^m$, some $u\in W$ has at least $\dim W$ nonzero coordinates. Choose $u$ with largest <support of a vector> $S$. Restriction from $W$ to $F^S$ is injective: a nonzero vector in its <kernel of a linear map> vanishes on $S$, so adding it to $u$ would enlarge $S$. Thus $\dim W\leq|S|$. The argument works over <finite fields>, where assuming a generic vector avoids finitely many hyperplanes would be invalid.
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