bilineaire

Bilinearity is a fundamental concept in mathematics, referring to functions that are linear in each of two arguments taken separately. A bilinear map is a function B : V × W → X between vector spaces (or modules) over a common field such that, for fixed w ∈ W, the map v ↦ B(v,w) is linear in V, and for fixed v ∈ V, the map w ↦ B(v,w) is linear in W. When V = W = X, the map is often called a bilinear form. Bilinear maps satisfy the identities B(av₁+bv₂,w) = a B(v₁,w)+b B(v₂,w) and B(v,aw₁+bw₂) = a B(v,w₁)+b B(v,w₂) for scalars a, b and vectors v, v₁, v₂, w, w₁, w₂.

A classic example is the dot product on ℝⁿ, defined by ⟨x,y⟩ = Σ x_i y_i, which is symmetric and positive‑definite. Another example is matrix multiplication viewed as a bilinear map (A,B) ↦ AB, linear in each factor when the other is held fixed. In functional analysis, the inner product on a Hilbert space is a bilinear (or sesquilinear) form, extending the notion of dot product to infinite dimensions.

Bilinear forms can be classified according to symmetry: a form B is symmetric if B(v,w)=B(w,v) and skew‑symmetric if B(v,w)=−B(w,v). Over fields of characteristic not equal to two, any bilinear form decomposes uniquely into a sum of symmetric and skew‑symmetric parts. The matrix representation of a bilinear form relative to a basis is a matrix M such that B(v,w)=vᵀMw.

Applications of bilinear maps appear in differential geometry (tensor products), physics (stress‑energy tensors), computer science (hash functions), and algebraic geometry (pairings on elliptic curves). The concept also underlies the definition of tensor products, as the universal bilinear map from V × W to V⊗W.