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Inner Product Proof - Verify on L2[-1,1]
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[QUOTE="ElijahRockers, post: 4999934, member: 321900"] [h2]Homework Statement [/h2] This[B] [/B]question has two parts, and I did the first part already I think. If B = {u1, u2, ..., un} is a basis for [B]V[/B], and ##v = \sum_{i=1}^n a_i u_i## and ##w = \sum_{i=1}^n b_i u_i## Show ##<v,w> = \sum_{i=1}^n a_i b_i^* = b^{*T}a## Here's how I did it: ##<v,w> = <\sum_{i=1}^n a_i u_i , w> = \sum_{i=1}^n a_i<u_i , w>## ## = \sum_{i=1}^n a_i b_i^* <u_i , u_i> = \sum_{i=1}^n a_i b_i^*## Thus proved... however in class he mentioned ##a_i = <v,u_i >## for doing this but I'm not sure how... I've tried to examine it but I can't seem to justify it. And I think I did the proof without that, since <v,aw> = a*<v,w> Second part of the question, where I'm confused, is, verbatim: Verify ##V = L^2 [-1,1]##, where B is the set of orthonormal Legendre polynomials, ##p_0 (x) = \frac{1}{\sqrt{2}}## ##p_1 (x) = \sqrt{\frac{3}{2}}x## ##p_2 (x) = \sqrt{\frac{5}{8}}(3x^2 -1)## and [B]v,w[/B] are replaced by ##x-x^2## and ##12+x-3x^2## [h2]Homework Equations[/h2][h2]The Attempt at a Solution[/h2] Not really sure where to start... he mentioned a_i = <v, u_i > in class but I don't really feel comfortable with using that here because I don't understand how that's true. ( I feel like it's really simple, and that's why it's bothering me so much) If somebody could point me in the right direction as to why that expression is true, I could probably finish the question.. [/QUOTE]
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Inner Product Proof - Verify on L2[-1,1]
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