Let u x1 x2 and v y1 y2 which of the following are bilinea

Let u = [x_1 x_2] and v = [y_1 y_2], which of the following are bilinear forms on R^2? (a) f(u, v) = 2x_1 y_2 - 3x_2y_1 (b) f(u, v) = x_1 + y_2 (c) f(u, v) = 0 (d) f(u, v) = 1.

Solution

As per definition, a symmetric bilinear form f on a vector space is a bilinear map from two vectors in the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map i.e. f(u,v) = f(v,u), f(u+v,w) = f(u,w)+f(v,w) and f(ku,v) = kf(u,v) for all all the scalars k and all the vectors u,v in the vector space.

(a) f(u,v) = 2x₁y₂ -3x₂ y₁ and f(v,u) = 2y₁x₂-3y₂x₁ f(u,v) so that f is not a bilinear form.

(b) f(u,v) = x₁+y₂ and f(v,u) = y₁+x₂ f(u,v) so that f is not a bilinear form.

(c) f(u,v) = 0 and f(v,u) = 0 so that f(u,v) = f(v,u). Further, f(u+v,w) =0 = f(u,w)+f(v,w). Also, f(ku,v) 0= kf(u,v). Hence f is a bilinear form.

(d) f(u,v) = 1 and f(v,u) = 1 so that f(u,v) = f(v,u). However, f(u+v,w) =1 while f(u,w)+f(v,w) = 1+1 = 2, f(u+v,w) . Hence f is not a bilinear form.