= Orthogonal complement for a bilinear form
{title2=$W^\perp=\{v:B(w,v)=0\text{ for all }w\in W\}$}
= Bilinear orthogonal complement
{synonym}
For a <bilinear form> on a finite-dimensional <vector space>, the displayed right <bilinear orthogonal complement> is a subspace. Nondegeneracy identifies the ambient space with its dual and gives $\dim W^\perp=\dim V-\dim W$. If the form is symmetric or alternating, orthogonality is reciprocal and $(W^\perp)^\perp=W$. For a general nonsymmetric form, left and right complements must be distinguished. A restricted symmetric or alternating form has <radical of a bilinear form> $W\cap W^\perp$; quotienting by that <radical of a bilinear form> gives a <nondegenerate> induced form.
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