Orthogonality- Gram-Schmidt Process for Complex Sequences

[gtfitzpatrick]

Homework Statement

Consider L₂, the inner product space of the complex sequences x = (x_n) such that \sum 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 can't be orthogonal because if it was there wouldn't be a part b but i can't 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 remove 0 from 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 I am missing?

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

 

[mrbohn1]

First of all...in part (a) you write that the set of vectors is orthonormal. This is not true: if a set of vectors are not orthogonal, then they are certainly not orthonormal.

Secondly, you are confusing the instuction about removing 0 from the set: it means you should remove the 0 vector if it appears in the set, NOT remove any zero entries in the vectors.

 

[gtfitzpatrick]

thanks for the reply,your right they obviously aren't ON either but i think i have shown that they are not orthogonal, right?

in second part, so it only the 0 vector i remove but leave the 0 entries in the vectors, thanks a million.
It still doesn't seem to be working out.is there something else I am missing.
im try to find 3 new sequences say u1,u2 and u3 i start with letting
x = u1
then
u2 = y- [<y,u1>/||u1||²] u1

u3 = z- [<z,u1>/||u1||²] u1 - [<z,u2>/||u2||²] u2

am i doing this right,its not working out?
Thanks for the help