bilinearity

Bilinearity is a property of a map B that takes two inputs from vector spaces and produces an output in a third space, typically over a common field. Let V and W be vector spaces over a field F, and let U be another vector space. A function B: V × W → U is bilinear if it is linear in each argument separately: for all v1, v2 in V and w in W, B(v1 + v2, w) = B(v1, w) + B(v2, w); for all v in V and w1, w2 in W, B(v, w1 + w2) = B(v, w1) + B(v, w2). Equivalently, for all scalars α in F, B(α v, w) = α B(v, w) and B(v, α w) = α B(v, w). Bilinearity does not require joint linearity in the pair (v, w).

When the codomain U is the field F, B is called a bilinear form on V × W (and on V × V when W = V). Bilinear forms can be characterized further by symmetry (B(v, w) = B(w, v)) or skew-symmetry (B(v, w) = −B(w, v)).

Examples include: the standard dot product on R^n, which is bilinear; matrix multiplication viewed as a map from M_{m×n} × M_{n×p} to M_{m×p}, bilinear in each factor; the evaluation pairing V × V* → F, (v, f) ↦ f(v), bilinear; and the cross product in R^3, which is bilinear in its two vector arguments.

Bilinear maps relate to tensor products: every bilinear map V × W → U corresponds to a unique linear map V ⊗ W → U, and conversely, composing a linear map with the tensor product yields a bilinear map. This connection underpins many constructions in linear algebra and multilinear algebra.