- Name Please Answer Each Of The Questions Below Adhering To The Following Principle Although I Know That I Can Get H 1 (135.37 KiB) Viewed 25 times
Name: Please answer each of the questions below, adhering to the following principle: • Although I know that I can get help from others, the solutions that I am presenting represent my own understanding and work. Scan and upload your responses to Gradescope before noon on Thursday, May 5. Recall that if V is a vector space then an inner product on V is a function (-:-): V XV +R satisfying the properties: (a) (u.v) = (v. u) for any u and v in V. (b) (au + bv,w) = a(u, w) + b(v.w) for any scalars a, b and any vectors u.v.w in V. (c) (v.v) > 0 with equality only if v = 0. We call V an inner product space if V is a vector space with an inner product. In this case, we can define the norm (or length or magnitude) of a vector v in V to be ||0|| = Vu.u). Consider the set C[0, 1] of continuous functions on the interval [0, 1]. You may take as granted that C[0, 1] is a vector space. We define a map (:) : C[0, 1] x C[0,1] → R via (5.9) = ['s(e)gt) dt. 1. Show that (,), as defined above, is an inner product. Namely, show that (a) property (i) holds: (b) property (ii) holds: (c) property (iii) holds. You may take as granted any facts about integration from Calculus that may be needed. 2. Consider the vector space P3 = {a+bt + ct+dt | a,b,c are scalars). This is a subspace of C[0,1], hence via the inner product defined above, an inner product space. Let f(t)=1, f(t) = t; 83(t) = +", f(t) =ť. (a) Compute the inner products (f. Sj) for each i = 1,2,3,4 and j = 1,2,3,4. In other words fill out the table: fi 12 fi 12 f where the entry in the i-th row and j-column is (fy). In the table you are only asked to compute (f) when i 3 j. Why is it not necessary to also compute (S., ) if i > j? (b) Are any of the vectors in S = {f1.12. f3, fa} unit vectors? Which, if any, of the vectors in S are orthogonal to each other? (c) Let 91 = ſı. Now find a nonzero vector 92 € span(f1, f2) = P, which is orthogonal to 91. Hint: You may find it helpful to use property (ii) and your table from part (b) so that you don't have to compute any more integrals directly. (d) Now find a nonzero vector 93 € span(f1. 12. /3) = P2 that is orthogonal to both g and 92 (e) Find a nonzero vector 94 € span(f1, 12, 13, SA) = Pthat is orthogonal to 91 92 and 93- (f) For each i = 1, 2, 3, 4, let hi = 19.01 What does the analogous table to that you computed in part (a) look like for the set {h1, h2, h3, h4}? 9