Degenerate form - Citizendia

Definition

In mathematics, specifically linear algebra, a degenerate bilinear form $f (x, y)$ on a vector space V is one such that the map from $V$ to $V *$ (the dual space of $V$ ) given by $v \mapsto f(-,v)$ is not an isomorphism. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Linear algebra is the branch of Mathematics concerned with In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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

$f(x,y)=0\,$ for all $y \in V.$

A nondegenerate form is one that is not degenerate, meaning that $v \mapsto f(-,v)$ is an isomorphism, or equivalently in finite dimensions, if and only if

$f(x,y)=0\,$ for all $y \in V$ implies that x = 0. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

There is the closely related notion of a perfect pairing; these agree over fields but not over general rings. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where

The most important examples of nondegenerate forms are inner products and symplectic forms. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the

Infinite dimensions

Note that in an infinite dimensional space, we can have a bilinear form f for which $v \mapsto f(-,v)$ is injective but not surjective (confer Hilbert's paradox). In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every Hilbert's paradox of the Grand Hotel is a mathematical Paradox about Infinite sets presented by German mathematician David Hilbert (1862–1943 For example, on the space of continuous functions on a closed bounded interval, the form

$f(\phi,\psi) = \int\psi(x)\phi(x) dx$

is not surjective: for instance, the Dirac delta functional is in the dual space but not of the required form. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. On the other hand, this bilinear form satisfies

$f(\phi,\psi)=0\,$ for all $\phi\,$ implies that $\psi=0.\,$

Terminology

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

$\{x\in V \mid f(x,y) = 0 \mbox{ for all } y \in V\}$

forms a totally degenerate subspace of V. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. f is nondegenerate if and only if this subspace is trivial. ↔

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. In Mathematics, the dimension of a Vector space V is the cardinality (i Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Likewise, a nondegenerate form is one for which the associated matrix is non-singular. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- These statements are independent of the chosen basis.

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. Nondegenerate bilinear forms are also sometimes called perfect pairings.

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