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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 6.): V XV R satisfying the properties: (a) (u.v) = (v.u) for any u and v in V. (b) (au + bu, w) = a(u, w) + b(u.w) for any scalars a, b and any vectors u,v.w in V. (c) (u, 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|| = (v.v). Consider the set C[0, 1] of continuous functions f 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] < C[0, 1] + R via (1.9) = ſ f(t)g(t) 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. f(t) = 1, 3(t) = 13. (a) Compute the inner products (fi) for each i = 1.2.3, 4 and j = 1,2,3,4. In other words fill out the table: fi 12 fs f4 fi 12 fs f4 where the entry in the i-th row and j-column is (../;). In the table you are only asked to compute (SS) when i <j. Why is it not necessary to also compute (Sy) if i > j? (b) Are any of the vectors in S = {f1, f2. f3: fa} unit vectors? Which, if any, of the vectors in S are orthogonal to each other? (c) Let 91 = f. Now find a nonzero vector 92 € span(f1, 12) = 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, 52, 53) = P, that is orthogonal to both gi and 92: (e) Find a nonzero vector ga Espan(f1, 12, 13, Sa) = P3 that is orthogonal to 91 92 and 93- (f) For each i = 1, 2, 3, 4, let h = What does the analogous table to that you computed in part (a) look like for the set {hu, h2, h3, h4}?