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...

In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements...

varieties Degenerate bilinear form, a bilinear form on a vector space V whose induced map from V to the dual space of V is not an isomorphism Degenerate distribution...

In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the...

realized as linear functionals on spaces of test functions. Every non-degenerate bilinear form on a finite-dimensional vector space V induces an isomorphism V...

Thus, bq is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form q ( x ) = b ( x , x ) ,...

quadratic form is non-degenerate and has the signature (a, b), then its isotropy index is the minimum of a and b. An important example of an isotropic form over...

vector field X at 0 is given by the signature of a certain non-degenerate bilinear form (to be defined below) on the local algebra BX. The dimension of...

finite-dimensional vector space V {\displaystyle V} endowed with a non-degenerate bilinear form ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is canonically...

the degenerate bilinear form d ( v , w ) = v 1 w 1 + v 2 w 2 + v 3 w 3 . {\displaystyle d(v,w)=v_{1}w_{1}+v_{2}w_{2}+v_{3}w_{3}.} This degenerate scalar...

non-degenerate bilinear form on TpM by means of g S ( X p , Y p ) = [ S X p , Y p ] . {\displaystyle g_{S}(X_{p},Y_{p})=[SX_{p},Y_{p}]\,.} This bilinear form...

inner product (such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism V → V ∗ , {\displaystyle V\to V^{*},}...

anticommutation relations in terms of a symplectic and a symmetric non-degenerate bilinear form. In addition, the binary elements in this graded Weyl algebra give...

a symplectic bilinear form. A symplectic bilinear form is a mapping ω : V × V → F {\displaystyle \omega :V\times V\to F} that is Bilinear Linear in each...

represent 4 reflections. Let (V, b) be an n-dimensional, non-degenerate symmetric bilinear space over a field with characteristic not equal to 2. Then...

is a 4-dimensional real vector space equipped with a non-degenerate, symmetric bilinear form on the tangent space at each point in spacetime, here simply...

3D4. A duality between two vector spaces over a field F is a non-degenerate bilinear form V 1 × V 2 → F , {\displaystyle V_{1}\times V_{2}\to F,} i.e., for...

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and...

\mathbf {Y} )} is called the Killing form; it is used to classify Lie algebras. The trace defines a bilinear form: ( X , Y ) ↦ tr ⁡ ( X Y )   . {\displaystyle...

non-degenerate invariant symmetric bilinear form. However the converse is false: a Lie algebra with a non-degenerate invariant symmetric bilinear form need...

easy and standard (uses the fact that the trace defines a non-degenerate bilinear form). Let A be a finitely generated algebra over a field k that is...

\mathrm {Vir} } , so V {\displaystyle V} is equipped with a non-degenerate bilinear form ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} and there is an algebra...

finite-dimensional. In this case, such an isomorphism is equivalent to a non-degenerate bilinear form φ : V × V → K {\displaystyle \varphi :V\times V\to K} In this case...

bilinear complexity or rank of a bilinear map is an important concept in the asymptotic complexity of matrix multiplication. The rank of a bilinear map...

follows. Given a module M over a *-ring R, let B(M) be the space of bilinear forms on M, and let T : B(M) → B(M) be the "conjugate transpose" involution...

sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a positive-definite symmetric bilinear form. The binomial expansion...

generally, given a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field, the orthogonal group of the form is the group of...

homeomorphicPages displaying short descriptions of redirect targets Degenerate bilinear form – Concept in linear algebra Dual system – Dual pair of vector spaces...

the point p {\displaystyle p} . A metric tensor is a non-degenerate, smooth, symmetric, bilinear map that assigns a real number to pairs of tangent vectors...

symmetric bilinear form on H2k(M,R); and therefore to a quadratic form Q. The form Q is non-degenerate due to Poincaré duality, as it pairs non-degenerately with...