Question

Section 5.5 Orthonormal Sets: Problem 6 Previous Problem Problem List Next Problem 1 (1 point) Use the inner product < f, g >= . f(x)g(x)dx in the vector space C°[0, 1] to find the orthogonal projection of f(x) = 6x2 + 1 onto the subspace V spanned by g(x) = x - and h(x) = 1. projy(f) =

Answer 1

Given that <f, g> = St143 g(4) dx basis of M. 2-1 Ž. h(x) = 1 g101) = here can see that <g, h> = s' - £) (1) de [ - 4*]. - cgins g (2) and head are orthogonal. Theretone , {x-1/2 '} es an orthogonal basis of webspace v. Now, fire) = 62? +1 Then orthogonal projection of t12) onto the subspace v h Proj, (f) - <fig> .gt <g, g> <fih> ch.h> CS Scanned with CamScanner

< f g > so (x-4) (6**+1) dre = [(x - 3****-432 1 - [ Sony - XS+ 2 = old 2 1 3/2 - 1 + 1 Ź - <9, g) = S(x-Ź) de = {(x²-x + & Jare [ - + -] Ź - { + 5 tt s, (6**+1) dre [ 24° +x].' U 12 <f, h) = 3 <h h> = s' due 1 Hence Proj v ft) = 6(4-5) +3(1) 64-3+3 = 60 Proj, (f) S Scannecktritt CamScanner