Abstract Linear Algebra - Math 416 Spring 2015 Homework 11 1. Let V be an inner product space, and let x, y V . Show that if x y, then x+y 2 = x 2 + y 2 . 2. For each of the following inner products, write down an orthonormal basis for P3 (R) using the Gram-Schmidt Algorithm applied to the basis (1, x, x2 , x3 ). (a) f, g = (b) f, g = (c) f, g = 1 0 f (t)g(t) dt, 2 0 f (t)g(t) dt, 2 1 f (t)g(t) dt, 3. Let be a basis for a subspace W of an inner product space V , and let z V . Show that z W if and only if z, v = 0 for all v . 4. Let V be a nite-dimensional vector space, and S a nonempty subset of V . Show that S is a subspace of V . 5. Let V be a vector space, and W a nite-dimesional subspace. Show that (W ) = W . 6. Let S be any nite subset of V . Show that Span(S) = (S ) . 7. Let = (v1 , . . . , vn ) be an orthonormal basis for V . Show that for any x, y V , n x, y = x, vi y, vi . i=1 8. Recall homework problem 1.10. Recall that we say a function is even if x, f (x) = f (x) and odd if x, f (x) = f (x). Show that W1 corresponds to the set of odd polynomials and W2 the set of even polynomials. Show that W2 = W1 and W1 = W2 . 9. Let T be a linear operator on an inner product space V . Let U1 = T + T and U2 = T T . Show that U1 , U2 are both self-adjoint. 10. Let V be an inner product space, and choose y, z V . Dene T : V V by T (x) = x, y z. Show that T is linear. Then show that T exists, and compute an explicit formula for it. Compute the rank of T and T