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Any bilinear map is a multilinear map. For example, any inner product on a -vector space is a multilinear map, as is the cross product of vectors in .; The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
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 each non-zero vector v in one of the two vector spaces, the pairing with v is a non-zero linear functional on the other.
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 of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:
Between trilinear coordinates and distances from sidelines [ edit ] For any choice of trilinear coordinates x : y : z to locate a point, the actual distances of the point from the sidelines are given by a' = kx , b' = ky , c' = kz where k can be determined by the formula k = 2 Δ a x + b y + c z {\displaystyle k={\tfrac {2\Delta }{ax+by+cz ...
In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × V → F. If V is a vector space with dual space V ∗, then the canonical evaluation map, b(f, v) = f(v) is a bilinear map from V ∗ × V to the base field. Let V and W be vector spaces over the same base field F.
Trilinear interpolation is the extension of linear interpolation, which operates in spaces with dimension =, and bilinear interpolation, which operates with dimension =, to dimension =. These interpolation schemes all use polynomials of order 1, giving an accuracy of order 2, and it requires 2 D = 8 {\displaystyle 2^{D}=8} adjacent pre-defined ...
Bilinear and trilinear interpolation, using multivariate polynomials with two or three variables; Zhegalkin polynomial, a multilinear polynomial over ; Multilinear form and multilinear map, multilinear functions that are strictly linear (not affine) in each variable; Linear form, a multivariate linear function
Sylvester's law of inertia is a theorem that asserts that the numbers of 1 and of –1 depends only on the bilinear form, and not of the change of basis. Symmetric bilinear forms over the reals are often encountered in geometry and physics, typically in the study of quadrics and of the inertia of a rigid body.