Results 1 to 8 of 8

Math Help - Orthogonalization proof

  1. #1
    Junior Member Greg98's Avatar
    Joined
    Oct 2009
    From
    Brugge, BEL
    Posts
    39

    Orthogonalization proof

    Hello,
    the task is following. Let's assume that S=\{y_1,y_2,y_3\} is orthogonal vector set of subspace V and \vec{0} \notin S. Also x \in V. It's defined that:
    y=x-\sum_{i=1}^{3}\frac{(x, y_i)}{(y_i, y_i)}y_i

    So, the problem is to show that y is orthogonal to all vectors of S.

    When the sum is opened, we get:
    y=x-\frac{(x, y_1)}{(y_1, y_1)}y_1-\frac{(x, y_2)}{(y_2, y_2)}y_2-\frac{(x, y_3)}{(y_3, y_3)}y_3

    It seems like formula from Gram-Schmidt process, so in my opinion it must be y \perp x, \forall x \in S, because G-S process orthogonalizes vectors. I think I need a little bit more evidence to show the proposition, but I don't know how to continue. Any help is welcome. Thanks!
    Last edited by Greg98; November 30th 2010 at 07:42 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    If V is a subspace, you absolutely must have \vec{0}\in V. Are you sure you meant \vec{0}\not\in V?

    Also, did you mean that S is an orthogonal basis for V?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member Greg98's Avatar
    Joined
    Oct 2009
    From
    Brugge, BEL
    Posts
    39
    Quote Originally Posted by Ackbeet View Post
    If V is a subspace, you absolutely must have \vec{0}\in V. Are you sure you meant \vec{0}\not\in V?
    Oops... I meant \vec{0}\not\in S. Does that make sense?

    Quote Originally Posted by Ackbeet View Post
    Also, did you mean that S is an orthogonal basis for V?
    Yeah, that's what I'm trying to say.

    Sorry about ambiguity!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Ok, so \vec{0}\not\in S, but you have to be able to get \vec{0} from a linear combination of vectors in S, right?

    Moving on to the proof, why not compute

    (y_{k},y)?

    I think you'll only need to do one - the other two are analogous. What should you get as the result of this computation?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member Greg98's Avatar
    Joined
    Oct 2009
    From
    Brugge, BEL
    Posts
    39
    Quote Originally Posted by Ackbeet View Post
    (y_{k},y)?

    I think you'll only need to do one - the other two are analogous. What should you get as the result of this computation?
    I think I should get \vec{0}, because they must be orthogonal. I know how the dot product is defined, but in this case I'm puzzled, because I don't know how many elements there's in y_{k} etc.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Well, you'll get the scalar 0, since the result of any dot product is a scalar, right? But yes, you should get zero if they are orthogonal. You can't compute the dot products directly, but you can use various theorems you know about dot products, such as:

    (x,\alpha y)=\alpha(x,y), for all scalars \alpha and vectors x and y.

    (x,y+z)=(x,y)+(x,z) for all vectors x, y, z.

    You also know that S is an orthogonal set (all the vectors in it are mutually orthogonal). That tells you something about, say, (y_{1},y_{2}), right?

    Just dive in and see what happens.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member Greg98's Avatar
    Joined
    Oct 2009
    From
    Brugge, BEL
    Posts
    39
    Got it now with using your help and proof of Gram-Schmidt process. Thanks very much!
    Follow Math Help Forum on Facebook and Google+

  8. #8
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    You're welcome. Have a good one!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Gram Schmidt Orthogonalization with functions
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 1st 2010, 05:40 PM
  2. Replies: 5
    Last Post: October 19th 2010, 10:50 AM
  3. Gram - Schmidt Orthogonalization
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 13th 2010, 04:16 AM
  4. Proof with algebra, and proof by induction (problems)
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: June 8th 2008, 01:20 PM
  5. proof that the proof that .999_ = 1 is not a proof (version)
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: April 14th 2008, 04:07 PM

Search Tags


/mathhelpforum @mathhelpforum