bilineaar

Bilinear, or bilineaar in some languages, describes a concept in linear algebra concerning maps that are linear in each argument separately. A bilinear map B takes two vectors from possibly different vector spaces, B: V × W → X, and satisfies linearity in both arguments: B(αv1 + βv2, w) = αB(v1, w) + βB(v2, w) and B(v, αw1 + βw2) = αB(v, w1) + βB(v, w2) for all vectors v, v1, v2 ∈ V; w, w1, w2 ∈ W and scalars α, β.

When V = W and the codomain is the field F, B is called a bilinear form on V. Such a form provides a scalar B(v, w) for any pair of vectors and is determined by a matrix representation relative to chosen bases: B(v, w) = v^T A w for some matrix A. The form is symmetric if B(v, w) = B(w, v) for all v, w, and alternating (or skew-symmetric) if B(v, v) = 0 for all v.

A related concept is the quadratic form Q(v) = B(v, v) when B is symmetric. The polarization identity often allows recovery of B from Q. In geometry and physics, bilinear forms underpin structures such as the metric tensor, which assigns a bilinear form to tangent spaces on manifolds, and thereby defines lengths and angles.

Examples include the standard dot product on R^n, which is a bilinear, symmetric form; other bilinear forms arise from choosing different bases or coordinate systems. Bilinear forms are foundational in defining orthogonality, projections, and dualities in linear algebra, as well as in tensor analysis and differential geometry.