Degenerate Form - Terminology

Terminology

If ƒ vanishes identically on all vectors it is said to be totally degenerate. Given any bilinear form ƒ on V the set of vectors

forms a totally degenerate subspace of V. The map ƒ is nondegenerate if and only if this subspace is trivial.

Sometimes the words anisotropic, isotropic and totally isotropic are used for nondegenerate, degenerate and totally degenerate respectively, although definitions of these latter words can vary slightly between authors.

Beware that a vector such that is called isotropic for the quadratic form associated with the bilinear form and the existence of isotropic lines does not imply that the form is degenerate.