Let V be a finite-dimensional vector space over a subfield F of the complex numbers, and let S be the set of all symmetric bilinear forms on V.
(a) Show that S is a subspace of L(V,V,F).
(b) Find dimS.
Let Q be the set of all quadratic forms on V.
(c) Show that Q is a subspace of the space of all functions from V into F.
(d) Describe explicitly an isomorphism T of Q onto S, without reference to a basis.
(e) Let U be a linear operator on V and q an element of Q. Show that the equation (U†q)(α)=q(Uα) defines a quadratic form U†q on V.
(f) If U is a linear operator on V, show that the function U† defined in part (e) is a linear operator on Q. Show that U† is invertible if and only if U is invertible.