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Homework Help: Quick question about notation (Linear Algebra)

  1. Aug 5, 2008 #1
    [tex]p\in P_3(\mathbb{F})[/tex]

    What does [tex]\overline{p(z)}[/tex] mean?

    I would guess that it's related to the complex conjugate, but I'm not sure. For context, I'm dealing with an inner product space defined by [tex]\langle p,q\rangle=\intop_{0}^{1}p(z)\overline{q(z)}dz[/tex]

    Thanks!
     
  2. jcsd
  3. Aug 5, 2008 #2

    Dick

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    Yes, it's a complex conjugate.
     
  4. Aug 6, 2008 #3

    HallsofIvy

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    that's an "inner product" of complex valued functions. The reason for the complex conjugate is so that
    [tex]<f(z),f(z)>= \int_0^1 f(z)\overline{f(z)}dz[/tex]
    will be a non-negative real number.
     
  5. Aug 6, 2008 #4
    Inner Product Space Norm

    Thanks for the help so far. I'm trying to find the norm for the this inner product. So far I have:

    [tex]\Vert p\Vert=\sqrt{\left\langle p,p\right\rangle }=\sqrt{\intop_{0}^{1}p(z)\overline{p(z)}dz}=\sqrt{\intop_{0}^{1}|p(z)|^{2}dz}[/tex]

    I also know that [tex]p(z)[/tex] can be denoted [tex]\sum_{i=0}^{n}a_{i}z^{i}[/tex] but I don't know how to connect the two beyond placing the sum within the absolute value.

    Is there a way to simplify [tex]|\sum_{i=0}^{n}a_{i}z^{i}|^{2}[/tex]?
     
  6. Aug 6, 2008 #5

    Dick

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    You've already written the norm of the inner product. Do you want to write it in terms of the a_i's? Is P_3 third degree polynomials? Or second? Either way it's going to be kind of uselessly complicated. If it's second, e.g. write (a0+a1*z+a2*z^2) times the complex conjugate and integrate over the real variable z in [0,1].
     
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