bilinearer

Bilinearer is a term used in some mathematical contexts to denote a bilinear map or bilinear form. In its core sense, a bilinearer is a function B that takes two vectors from potentially different vector spaces and returns a field element, being linear separately in each argument.

Formally, let V and W be vector spaces over a field F. A bilinearer is a function B: V × W → F that is linear in each argument. That is, for all v1, v2 in V, w in W and scalars α, β in F, B(α v1 + β v2, w) = α B(v1, w) + β B(v2, w); and B(v, α w1 + β w2) = α B(v, w1) + β B(v, w2). When V = W and the target is F, B is often called a bilinear form.

Examples include the dot product on R^n, defined by B(v, w) = v^T w, and matrix-based expressions such as B(v, w) = v^T M w for a fixed matrix M. Matrix multiplication itself can illustrate bilinearity in its two arguments when viewed as a map on appropriate spaces.

Key properties include symmetry (B(v, w) = B(w, v)) and nondegeneracy (if B(v, w) = 0 for all w implies v = 0). Bilinearers are closely related to quadratic forms via Q(v) = B(v, v). In finite dimensions, every bilinear form corresponds to a linear map under an appropriate inner product.

Note on terminology: bilinearer is not standard in all texts. The conventional terminology is bilinear form or bilinear map; bilinearer may appear as an interdisciplinary or historical variant in some sources.