(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider L_{2}, the inner product space of the complex sequencesx= (x_{n}) such that [tex]\sum[/tex] x_{i}converges,

with the inner product given by

<x,y> = (sum of) x_{i}y_{i}(complex conjugate)

Now let

x= (1,0,1,0,1,0,0,0...)

y= (1,i,0,i,0,i,0,0,0...)

z= (-1,1,i,-1,1,i,0,0..)

(x_{n}= y_{n}= z_{n}for all n>7 = 0)

a) Is the set {x,y,z} an orthogonal set in L2?

b) If not use the Gram-Schmidth orthogonalization process to get an orthogonal set with the same span

Sol.

A) well i know it cant be orthogonal because if it was there wouldn't be a part b but i cant give that as an answer so for them to be orthogonal <x,y> = 0 but i get <x,y> = 1 so they are not orthogonal(but are orthonormal) so the answer is no, the set is not an orthogonal set in L2

i think i've got that right?

b) to apply Gram-schmidth you first have to remove0from the list

which i did giving

x = 1,1,1,0,0,0,0...

y = 1,i,i,i,0,0,0,0,..

z = -1,1,i,-1,1,i,0,0...

and then fill into the formula, but when i do this its not working out, is there something im missing?

Thanks a mill for reading and sorry some of the code didn't work,i hope you can understand the question.

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# Homework Help: Orthogonality- Gram-Schmidt

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