Orthogonal complement for a bilinear form ($W^{\perp }=\{v:B(w,v)=0\text{ for all }w\in W\}$)

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.