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Inner Product

  1. Sep 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Let v,w be vectors in a complex inner product space such that ||v|| = 1,
    ||w|| = 3 and <v,w> = 1 + 2i. Find ||v + iw||.


    2. Relevant equations
    The properties of an inner product.


    3. The attempt at a solution
    I figured
    [tex]||v+iw||^2[/tex] = <v+iw,v+iw>

    Then using the properties of the inner product, I broke it up;

    [tex]||v+iw||^2[/tex] = <v,v+iw>+<iw,v+iw>

    and <v,v+iw> = <v,v> + <v,iw> and using the 'conjugate symmetry'
    = <v,v> - i<v,w>
    = 1-i(1+2i)
    = 3-i

    Now for <iw,v+iw>;
    =i<w,v+iw>
    =i{<w,v>+<w,iw>}
    =i{1-2i - i<w,w>} (since <v,w>=[tex]\overline{<w,v>}[/tex])
    =i+2-9i
    =2-8i

    Now combining it all i get [tex]||v+iw||^2[/tex] = 5-9i

    But this is supposed to be the square of the magnitude of v+iw.. so it should be a real number right?

    Can somebody help point out where I have gone wrong?
    Cheers.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 11, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Here is your error. i(1- 2i- 9i)= i+ 2+ 9. You forgot the second i on the last term.

     
  4. Sep 11, 2009 #3
    oops, thanks a lot halls!
     
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