Bilineaarset

Bilineaarset refers to mathematical objects that are bilinear in two arguments. In linear algebra, a bilinear map is a function B: V × W → X that is linear in each argument separately. This means B(av1 + bv2, w) = aB(v1, w) + bB(v2, w) and B(v, aw1 + bw2) = aB(v, w1) + bB(v, w2) for all vectors v, v1, v2 in V, w, w1, w2 in W and scalars a, b.

When the codomain X is the underlying field F, a bilinear map on a single vector space V is called a bilinear form. A bilinear form B: V × V → F is symmetric if B(v, w) = B(w, v) for all v, w, and alternating if B(v, v) = 0 for all v. A form is nondegenerate if the only v with B(v, w) = 0 for all w is v = 0; this property yields useful duality relations.

Forms are commonly represented by matrices. Given a basis of V, B(v, w) can be written as v^T A w, where A is the Gram matrix of B. Changing basis alters A by a congruence transformation, preserving the bilinear structure. The dot product is a key example: <x, y> = x^T y is a symmetric, positive-definite bilinear form on R^n. Other examples include the Minkowski metric in relativity and various bilinear forms used in geometry and physics.

Bilinear maps relate to the tensor product: any bilinear map V × W → X corresponds to a linear map V ⊗ W → X. This connection underpins many constructions in mathematics and theoretical physics.