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Math Help - othogonal projections - question about definition

  1. #1
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    othogonal projections - question about definition

    I am a little bit confused by a wording on the orthogonal projection given in my study guide for linear algebra (LSE).

    It says "it is the projection onto S parallel to the orthongonal complement of S (for a particular subspace of S)."

    I am confused by the world 'parallel'. If we take a simple case of a plane S and its orthogonal complement So (a line perpendicular to the plane), then any vector x can be described as a sum of two projections: x=x1+x2, where x1 - is the projection of vector x onto a plane S, and x2 - the projection of vector x onto the line So. The projection of the vector onto a plane S - x1 - is not going to be parallel to the line So - since it will belong to the plane S, it will be perpendicular to So.

    So, looking back at the definition, what is 'parallel' in my example?

    thanks a lot! (sorry I cannot figure out how to cut and paste math signs into the message text).
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  2. #2
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    Quote Originally Posted by Volga View Post
    (sorry I cannot figure out how to cut and paste math signs into the message text).
    Why cut and past when you can learn to use LaTex.
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  3. #3
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    I think the narrative description was not too vague and someone can comment on the main question and not the side note in the parenthesis.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Volga View Post
    So, looking back at the definition, what is 'parallel' in my example?
    I understand. Choose for example in the usual euclidean vector space \mathbb{R}^3:

    S \equiv z=0. Then S_0 \equiv x=0 , y=0

    Choose v=(1,1,1). We have the decomposition:

    (1,1,1)=(1,1,0)+(0,0,1),\quad x_1=(1,1,0)\in S,\;x_2=(0,0,1)\in S_0

    and x_1 is the orthogonal projection of v onto S, and of course, x_1 is not parallel to S_0.

    Now choose for example P(2,3,5), the projection onto S of the point P parallel to the orthogonal complement of S is the intersection of P+\lambda (0,0,1) (parallel affine variety to S_0 through P) with S. In this case we obtain the point (2,3,0)

    Fernando Revilla
    Last edited by FernandoRevilla; December 23rd 2010 at 04:48 AM.
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  5. #5
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    Sorry Fernando it seems that your message got cut off in the end
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Volga View Post
    Sorry Fernando it seems that your message got cut off in the end
    I don't see it. Anyway, ask just what you want.

    Fernando Revilla
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  7. #7
    A Plied Mathematician
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    Fernando, I think Volga was referring to the fact that your last phrase in post # 4 is an incomplete sentence. At least, there's no verb there. It reads, "In this case the point (2,3,0)", which is incomplete.
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Ackbeet View Post
    Fernando, I think Volga was referring to the fact that your last phrase in post # 4 is an incomplete sentence. At least, there's no verb there. It reads, "In this case the point (2,3,0)", which is incomplete.
    All right!. ( ). I've corrected it.

    Thank you.

    Fernando Revilla
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  9. #9
    A Plied Mathematician
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    You're welcome.

    Cheers.
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  10. #10
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    I think I got it. Projection is not a vector, it is a function (mapping) so by drawing a mapping line parallel to So we obtain (2,3,0) - projection of (2,3,5) onto S.
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  11. #11
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Volga View Post
    ... so by drawing a mapping line parallel to So we obtain (2,3,0) - projection of (2,3,5) onto S.
    That's right.

    Fernando Revilla
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