A bilinear form on a real vector space is a function b:V×V->R that satisfies the following axioms for any scalar α and any choice of vectors v, w, v_1, v_2, w_1, and w_2. 1.b(α v, w) = b(v, α w) = α b(v, w) 2.b(v_1 + v_2, w) = b(v_1, w) + b(v_2, w) 3.b(v, w_1 + w_2) = b(v, w_1) + b(v, w_2). For example, the function b((x_1, x_2), (y_1, y_2)) = x_1 y_2 + x_2 y_1 is a bilinear form on R^2.

bilinear function | multilinear form | symmetric bilinear form | vector space