Linear Algebra I Problem Set 5: Inner Products and Bilinear and Sesquilinear Forms Dr Nicholas Sedlmayr Friday February 19th 2016 Due: In class, February 26th 2016 1. Let V be the vector space over R of all polynomials of degree less than 3. I.e. V = {a0 + a1 x + a2 x2 : a0 , a1 , a2 , x R}. We can dene an inner product on this space as 1 f |g = dxf (x)g(x) . 0 1, x, x2 is an ordered basis of V, nd the corresponding matrix of inner products A. Use this matrix to calculate x + 1|x2 2 and x2 x + 5|x2 + 2x . Conrm the results by direct integration. (6) 2. Prove that the map F : V V C is a sesquilinear form on V = Cn where F (x, y) = xT Ay, A is an n n matrix, and x, y V . Furthermore T prove that if F is conjugate symmetric then A = A. (4) 3. A sesquilinear form on C3 is dened by x 0 1 2i 0 0 i y . F ((x, y, z)T , (x , y , z )T ) = x y z 1 + 2i 0 i 1 z What is the matrix of F with respect to the bases (a) {(i, 1, 0)T , (1, 1 + i, 1)T , (0, 0, i)T }, and (b) {(1 3i, 2 + i, 1 + 2i)T , (2 + i, 1 2i, 2 + i)T , (0, 0, i)T }? Is F an inner product? (5) 4. Ex. 4.8. An alternating form F is a bilinear form on a vector space V satisfying F (v, v) = 0 v v. (a) Show that if F is an alternating form then F (u, v) = F (v, u), i.e. that F is skew symmetric. (3) (b) Give an example of an alternating form (other than the zero function). (2) Total available marks: 20 1