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- Degenerate_form abstract "In mathematics, specifically linear algebra, a degenerate bilinear form ƒ(x,y) on a vector space V is one such that the map from to (the dual space of ) 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 A nondegenerate or nonsingular form is one that is not degenerate, meaning that is an isomorphism, or equivalently in finite dimensions, if and only iffor all implies that x = 0.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.There is the closely related notion of a unimodular form and a perfect pairing; these agree over fields but not over general rings.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.".
- Degenerate_form wikiPageID "772150".
- Degenerate_form wikiPageRevisionID "546859172".
- Degenerate_form hasPhotoCollection Degenerate_form.
- Degenerate_form subject Category:Bilinear_forms.
- Degenerate_form type Abstraction100002137.
- Degenerate_form type BilinearForms.
- Degenerate_form type Form106290637.
- Degenerate_form type LanguageUnit106284225.
- Degenerate_form type Part113809207.
- Degenerate_form type Relation100031921.
- Degenerate_form type Word106286395.
- Degenerate_form comment "In mathematics, specifically linear algebra, a degenerate bilinear form ƒ(x,y) on a vector space V is one such that the map from to (the dual space of ) given by is not an isomorphism.".
- Degenerate_form label "Degenerate form".
- Degenerate_form sameAs m.03b7fr.
- Degenerate_form sameAs Q8214677.
- Degenerate_form sameAs Q8214677.
- Degenerate_form sameAs Degenerate_form.
- Degenerate_form wasDerivedFrom Degenerate_form?oldid=546859172.
- Degenerate_form isPrimaryTopicOf Degenerate_form.