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Simple inner product question on CMOplex SPace

  1. Apr 26, 2013 #1
    Let the inner product on P2(C) be defined as:

    ∫ (from -1 to 1) p(t)(conjugate) q(t) dt.

    using the gram schmidt process and the standard basis {1, x, x2} find an orthonormal basis.


    So my only issue really is interpreting this integral. I just wanted to test if the vectors in the basis were not orthogonal for my own assurance, but now I'm not sure if that is the case. So taking the inner product of < x , 1>:

    < x, 1> = x2/2 ? is that right? I mean there are no complex values in the standard basis so I don't see how the conjugate is changing anything.
     
  2. jcsd
  3. Apr 26, 2013 #2

    Dick

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    Yes, the conjugate isn't changing anything. NO, <x,1> is NOT x^2/2. It's a definite integral.
     
  4. Apr 26, 2013 #3


    Right, but now this is a past exam question I'm working on, and if that is the case. The set of standard basis vectors would each be orthogonal to each other. Which makes sense, then all I would have to do is "normalize" them to make them orthonormal. But that is only 3-4 lines at most. I don't know something seems odd about that.
     
  5. Apr 26, 2013 #4

    Dick

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    <1,x^2> isn't 0. What is it?
     
  6. Apr 26, 2013 #5


    True, thank god. I'm having another small calculation question on another inner product deinfed as:

    <p(x),q(x)> = P(0)(conjugate) q(0) + P(i)(conjugate) q(i)

    now the question is asking to find an orthonormal basis for P1(C)

    going through the gram schmidt again, I'm getting tangled up in an inner product that isn't giving me what I think it should.

    So using the standard basis of P1: {1,x} I let "1" be my initial vector. So to find the next vector call it v2:

    v2 = x - < x, 1>/<1,1> so I'm having issues with < x , 1 >.

    Doesn't < x, 1> = (-i)/2 ? thus giving me a basis vector of (x + i/2)....this didn't turn out being orthogonal
     
  7. Apr 26, 2013 #6

    Dick

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    Be careful with the order of the arguments to the inner product. <x,1>=i. <1,x>=(-i). Try it carefully. And maybe I have that wrong. Which function is the conjugate on, p or q?
     
  8. Apr 26, 2013 #7


    Great. That means the way I had accepted of finding ortogonal bases over the complex is flawed. I had worked the general form as being <v2, u1>/<u1, u1> because I had an issue with this before but over the regularly defined complex inner product and that brought about the right solution, the textbook had it defined as <u1,v2>/<u1,u1> but then doing it that way I didn't get the right solution.

    Looks like I'm going to have to try both and see which satisfies it in the proper way.
     
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