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Homework Help: An inner product problem

  1. Oct 5, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove that
    [tex]\left\langle\alpha x,y\right\rangle-\alpha\left\langle x,y\right\rangle=0[/tex] for [tex] \alpha=i [/tex]
    where
    [tex] \left\langle x,y\right\rangle=\frac{1}{4}\left\{\left\|x+y\right\|^{2}-\left\|x-y\right\|^{2}+i\left\|x+iy\right\|^{2}-i\left\|x-iy\right\|^{2}\right\}[/tex]


    2. Relevant equations


    3. The attempt at a solution
    I put the alpha*x into that equation and substract it from [tex]\alpha\left\langle x,y\right\rangle[/tex]
    unfortunately, I couldn't find zero, and what it yielded is
    [tex]\frac{1}{2}\left[\left\|x-y\right\|^{2}-\left\|x+y\right\|^{2}+\left\|x+iy\right\|^{2}-\left\|x-iy\right\|^{2}\right][/tex]

    How on earth can this expression yield zero?
     
  2. jcsd
  3. Oct 5, 2008 #2

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    I think you're slipping up somewhere. Maybe everything will be easier to manage if you rewrite the equation for [itex]\langle x,y \rangle[/itex] as:

    [tex]\langle x,y \rangle = \frac{1}{4} \sum_{k=0}^3 i^k \|x+i^ky\|.[/tex]
     
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