Math 118 Midterm Solutions: Convergence, Orthonormal Basis, Filters, Exams of Mathematics

Download Math 118 Midterm Solutions: Convergence, Orthonormal Basis, Filters and more Exams Mathematics in PDF only on Docsity! Answer only 4 of the following 5 questions. Indicate clearly which question you don’t want graded (or I’ll only grade the first four). Math 118 Midterm Spring 2009, Wilkening Name: 1. Let fn(x) =  0, 0 ≤ x ≤ 1√ n, 1 < x < 1 + n−1 0, 1 + n−1 ≤ x ≤ 3  . Answer the following questions true orfalse and give a short proof justifying each answer. a. (2.5 points) fn → 0 pointwise on [0, 3]. b. (2.5 points) fn → 0 uniformly on [0, 3]. c. (2.5 points) fn → 0 in L1[0, 3]. d. (2.5 points) fn → 0 in L2[0, 3]. 2. Let V be the set of vectors in C2 with inner product 〈 x, y 〉 = xT M y, M = ( 1 1 1 2 ) . a. (4 points) Find an orthonormal basis for V . b. (3 points) Let u = ( 1 1 ) , w = ( 1 0 ) and W = span{w} = {αw : α ∈ C}. Find the closest point v ∈ W to u in the norm of V , i.e. minimize ‖v − u‖2 = 〈v − u, v − u〉. c. (3 points) Let A = ( 1 −i 1 0 ) . Treating A as a linear operator A : V → V , find the adjoint matrix A∗. In case it is helpful, M−1 = ( 2 −1 −1 1 ) .