In mathematics, specifically linear algebra, a degenerate bilinear form f (x, y ) on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V ) given by v ↦ (x ↦ f (x, v )) is not an isomorphism.
Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map
There is the closely related notion of a unimodular form and a perfect pairing; these agree over fields but not over general rings.
Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map
For example, on the space of continuous functions on a closed bounded interval, the form is not surjective: for instance, the Dirac delta functional is in the dual space but not of the required form.
Geometrically, an isotropic line of the quadratic form corresponds to a point of the associated quadric hypersurface in projective space.
Such a line is additionally isotropic for the bilinear form if and only if the corresponding point is a singularity.