Bilineært

Bilineært (from the term bilinear) describes a property of functions or relations that are linear in each of two arguments when the other is held fixed. In mathematics the notion appears most commonly as a bilinear map or a bilinear form. A map f: V × W → X between vector spaces is bilinear if for every fixed w in W the map v ↦ f(v,w) is linear in v, and for every fixed v in V the map w ↦ f(v,w) is linear in w. When V = W and X is the base field, such a map is called a bilinear form.

Bilinear forms can be represented by matrices: given a basis, B(v,w) = v^T A w for some matrix A. Important subclasses include symmetric and skew-symmetric bilinear forms, and nondegenerate forms that induce isomorphisms between a space and its dual. Bilinear maps are closely related to tensor products: each bilinear map V × W → X factors uniquely through a linear map V ⊗ W → X.

Outside pure algebra, bilinear concepts appear in analysis and applied fields. Bilinear interpolation is a common technique in image processing and numerical analysis for estimating values on a two-dimensional grid by linear interpolation first in one direction and then the other. In signal processing and control theory, bilinear systems are dynamical models with inputs appearing multiplicatively with states. The bilinear property provides a unifying framework for many linear-algebraic constructions and practical algorithms.