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Orthogonal projection

  1. Feb 17, 2012 #1
    I found a final answer online, but my vector is slightly different. I haven't been able to catch my mistake.

    I'm supposed to find the orthogonal projection of the given vector on the given subspace W of the the inner product space V.

    P1 has dimension 2 and basis = {1,x}.

    http://i111.photobucket.com/albums/n149/camarolt4z28/File2.png [Broken]
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Feb 17, 2012 #2

    Fredrik

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    I haven't thought it through to the end, but doesn't the formula you're trying to use for the orthogonal projection onto W require that you use an orthonormal basis for W? (Check what your book says. I was too lazy to look it up myself or think about it).
     
  4. Feb 17, 2012 #3
    I didn't normalize {1,x} for W. That's the problem. Good catch. Thanks.
     
  5. Feb 19, 2012 #4
    I normalized the basis {1,x} for W: {1, sqrt(3)x}

    I'm getting for the projection (29/6) + (15/2)x.
     
  6. Feb 19, 2012 #5

    vela

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    You normalized the basis vectors, but they're still not orthogonal.
     
  7. Feb 20, 2012 #6
    The orthogonality slipped my mind. I suppose I should use Gram-Schdmit for that.

    {1, x-(1/2)}
     
  8. Feb 20, 2012 #7
    I don't know why this problem is giving me trouble.

    I used Gram-Schmidt on the standard basis for P1 and got {1, x-(1/2)}. I normalized this basis for W and got

    u1 = 1
    u2 = sqrt(12)(x-(1/2)).
     
  9. Feb 20, 2012 #8

    vela

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    Looks good.
     
  10. Feb 20, 2012 #9
    When I use the projection formula I get x - (62/66).

    The answer I found has x - (13/3).
     
  11. Feb 20, 2012 #10

    vela

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    I get neither. Show us your calculations.
     
  12. Feb 20, 2012 #11
    http://i111.photobucket.com/albums/n149/camarolt4z28/File21.png [Broken]
     
    Last edited by a moderator: May 5, 2017
  13. Feb 20, 2012 #12

    vela

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    You just added incorrectly when evaluating the last integral. You should get ##1/\sqrt{12}##.
     
  14. Feb 20, 2012 #13
    Shoot. You're right. I should have had a +1/2, not -1/2. The last integral should actually be sqrt(12)/12. I do end up with x + 26/6.

    Thanks for the help! I'm all finished with this assignment. I'm graduating in May, so I'm counting down the weeks and assignments! Haha.
     
  15. Feb 20, 2012 #14

    vela

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    Ah, yes, it is x+13/3. I didn't actually multiply it out here.
     
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