MATH 4450 - HOME WORK 6

(1) Let V be an n-dimensional vector space over the field K and B be a basis for V . Let Bil(V × V, K) be the set of all bilinear maps on V × V to K. (a) Prove that there is an isomorphism Bil(V × V, K) → Matn×n (K). We proved this in class when V = Rn and B is the standard basis. As I mentioned then, the same proof goes through (almost) verbatim. So this exercise is intended to make sure that you understand the various concepts involved. So first define the map and then show it is injective and surjective. (b) Let V ∗ be the dual vector space and B be the dual basis with respect to B. Show that with respect to the bases B and B , there is an isomorphism Lin(V, V∗ ) → Matm×n (K). (c) Conclude now that there is an isomorphism Bil(V × V, K) → Lin(V, V∗ ). (2) Let f : R2 × R2 → R defined by f (x, y) = x1 y1 + x1 y2 + x2 y1 + x2 y2 where x = (x1 , x2 ) and y = (y1 , y2 ). Find the matrix A such that f (x, y) = xt Ay. Finally check that the matrix that you have obtained is indeed the correct one. (3) Let V be a vector space of dimension n with basis B and < , > be a scalar product on V . Recall that any such scalar product is a bilinear form. Show that the matrix associated to this scalar product is a symmetric matrix. (4) Let V be an n-dimensional vector space. Prove that there to every non-zero linear functional φ : V → K, we there is an (n − 1)-dimensional subspace Wφ ⊂ V such that ker φ = Wφ . Show that if ψ : V → K is another linear functional such that ker ψ = ker φ, then there exists a non-zero scalar such that φ = cψ.

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