In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V&hairsp;) given by is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero x in V such that for all Nondegenerate forms A nondegenerate or nonsingular form is a bilinear form that is not degenerate, meaning that is an isomorphism, or equivalently in finite dimensions, if and only if for all implies that . The most important examples of nondegenerate forms are inner products and symplectic forms. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map be an isomorphism, not positivity. For example, a manifold with an inner product structure on its tangent spaces is a Riemannian manifold, while relaxing this to a symmetric nondegenerate form yields a pseudo-Riemannian manifold. Using the determinant If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero – if and only if the matrix is singular, and accordingly degenerate forms are also called singular forms. Likewise, a nondegenerate form is one for which the associated matrix is non-singular, and accordingly nondegenerate forms are also referred to as non-singular forms. These statements are independent of the chosen basis. Related notions If for a quadratic form Q there is a non-zero vector v ∈ V such that Q(v) = 0, then Q is an isotropic quadratic form. If Q has the same sign for all non-zero vectors, it is a definite quadratic form or an anisotropic quadratic form. There is the closely related notion of a unimodular form and a perfect pairing; these agree over fields but not over general rings. 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 numbers, split-complex numbers, and dual numbers. For z = x + ε y, the dual number form is x2 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 products and symplectic forms. Symmetric nondegenerate forms are