Question

Problem 5. Given a vector space V, a bilinear form on V is a function f : V x V -->R satisfying the following four conditions: f(u, wf(ū, ) + f(7,i) for every u, õ, wE V. f(u,ū+ i) = f(u, u) + f(ū, w) for every ā, v, w E V. f(ku, kf (ū, v) for every ū, uE V and for every k E R f(u, ku) = kf(u, u) for every u,uE V and for every k E R. (a) Given two bilinear forms f,g : V x V -> R, define a new function (f + g) : V x V --> R by (f g)(, u) f(,u)g(v,u). Prove that f+ g is also a bilinear form (b) Given a bilinear form f : Vx V -> V and a real number c E R, define a new function (cf): V x V R by (cf)(v,u) = cf(v, u). Prove that cf is also a bilinear form. (c) Using the previous two parts, prove that the set B(V) of all bilinear forms on V is a vector space (d) Let V be finite-dimensional, with basis (Ti,... , Tn). Suppose that f, g E B(V) are such that for all i,jn, it is the case that f(v, Tj) = g(vi,T,). Prove that f = g. (e) Still assuming that V is finite-dimensional, with basis (Ti,..., Tn), prove that for every n x n matrix [aij=1 tere exists a unique fe B(V) such that f(Vi,U;) = aij for all i, j < n. (f) Using the previous two parts, conclude that, if dim(V) = n, then B(V) is isomorphic to R"xn. A =

Answer 1

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