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Orthogonality- Gram-Schmidt

  1. Apr 27, 2010 #1
    1. The problem statement, all variables and given/known data

    Consider L2, the inner product space of the complex sequences x = (xn) such that [tex]\sum[/tex] 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. jcsd
  3. Apr 27, 2010 #2
    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.
     
  4. Apr 27, 2010 #3
    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
     
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