# Orthogonality- Gram-Schmidt

1. Apr 27, 2010

### gtfitzpatrick

1. The problem statement, all variables and given/known data

Consider L2, the inner product space of the complex sequences x = (xn) such that $$\sum$$ xi converges,
with the inner product given by
<x,y> = (sum of) xi yi(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..)

(xn = yn = zn 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 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 im missing?

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

2. Apr 27, 2010

### 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.

3. Apr 27, 2010

### 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 doesnt seem to be working out.is there something else im 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