= Complex quadratic forms have a large isotropic subspace
{title2=$\dim E=n-\lceil r/2\rceil$}
A <quadratic form> of rank $r$ on $\mathbb C^n$ is congruent to $x_1^2+\cdots+x_r^2$. Pair its nonzero coordinates into vectors $e_{2j-1}+ie_{2j}$, and include the <radical of a bilinear form>. Their span is a <totally isotropic subspace> and has dimension $\lfloor r/2\rfloor+n-r=n-\lceil r/2\rceil\ge\lfloor n/2\rfloor$. The form vanishes on every vector of this subspace, not merely on its basis vectors.
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