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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.
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.
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.
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:
Trilinear interpolation as two bilinear interpolations followed by a linear interpolation. Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid . It approximates the value of a function at an intermediate point ( x , y , z ) {\displaystyle (x,y,z)} within the local axial rectangular prism linearly ...
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space.A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui-meaning "one and a ...
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