Editing Degenerate bilinear form (section)

==Examples==
The study of real, quadratic algebras shows the distinction between types of quadratic forms. The product ''zz''* is a quadratic form for each of the [[complex number]]s, [[split-complex number]]s, and [[dual number]]s. For ''z'' = ''x'' + &epsilon; ''y'', the dual number form is ''x''<sup>2</sup> which is a '''degenerate quadratic form'''. The split-complex case is an isotropic form, and the complex case is a definite form.

The most important examples of nondegenerate forms are [[inner product]]s and [[symplectic form]]s. [[Symmetric bilinear form|Symmetric]] nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map <math>V \to V^*</math> be an isomorphism, not positivity. For example, a [[manifold]] with an inner product structure on its [[tangent space]]s is a [[Riemannian manifold]], while relaxing this to a symmetric nondegenerate form yields a [[pseudo-Riemannian manifold]].