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Gram-Schmidt orthonormal basis

  1. May 4, 2009 #1
    1. The problem statement, all variables and given/known data
    Let R^3 have the inner product <u, v> = u1v1 + 2u2v2 + 3u3v3. Use the Gram-Schmidt process to convert u1=(1,1,1), u2 = (1,1,0), u3 = (1,0,0) into a normal orthonormal basis


    2. Relevant equations
    I know the process for the orthonomoral converasion. I have no problem with that. I hope I can skip typing those equations out.

    3. The attempt at a solution
    My confusion is how to use the equation for the dot product. I am thinking that I need to do the Gram-Schmidt process then somehow put the result into the dot product equation.

    If I phrased anything wrong or left out something pertinent I apologize this is my first post. So I tried to follow the rules.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 4, 2009 #2

    matt grime

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    If you know the formula, then what is the problem? You have the vectors and the inner product, so you just plug and chug.
    Let's start you off - we need to normalize u_1, i.e replace u_1 with

    u_1/ sqrt(<u_1,u_1>) = u_1/sqrt(6).

    Note that this is the least sensible way to do this. It would be better to use u_3 as the first element to orthonormalize, since that satisfies <u_3,u_3> = 1, i.e. it is already a unit vector. But I presume that the book/class wants you to go through them in order and normalize u_1 to get v_1, then to work out u_2 - <u_2,u_1> and normalize that, etc.
     
  4. May 4, 2009 #3
    Then maybe I am reading too far into. I am just not sure how the dot product equation in the beginning of question is utilized. As in after I convert [tex]\vec{u_1}[/tex], [tex]\vec{u_2}[/tex], [tex]\vec{u_3}[/tex]. So do I just plug in the new normalized [tex]\vec{u_1}[/tex], [tex]\vec{u_2}[/tex], [tex]\vec{u_3}[/tex] and then use [tex]\vec{u_1}[/tex] = [tex]\vec{v_1}[/tex], [tex]\vec{u_2}[/tex]=[tex]\vec{v_2}[/tex], [tex]\vec{u_3}[/tex]=[tex]\vec{v_3}[/tex] into <u, v> = u1v1 + 2u2v2 + 3u3v3.
     
  5. May 4, 2009 #4

    Cyosis

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    The general Gramm-Schmidt formula reads:

    [tex]v_j=a_j-\left( \frac{\langle a_j,v_1 \rangle}{\langle v_1,v_1 \rangle}v_1+\frac{\langle a_j,v_2 \rangle}{\langle v_2,v_2 \rangle}v_2+\ldots+\frac{\langle a_j,v_{j-1} \rangle}{\langle v_{j-1},v_{j-1} \rangle}v_{j-1} \right)[/tex].

    This should make it pretty obvious where to use the dot product, so I am curious what kind of formula were you thinking of?
     
  6. May 4, 2009 #5

    HallsofIvy

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    For example, to "normalize" the first vector, you need to divide by its length. And its length is [itex]||v||= \sqrt{v\cdot v}[/itex]. Because of the definition of dot product, [itex](1,1,1)\cdot(1,1,1)= 1(1)+ 2(1)(1)+ 3(1)(1)= 6[/itex]. That's where matt grime got the [itex]\sqrt{6}[/itex] he divided the first vector by.
     
  7. May 4, 2009 #6
    Ok, I was forgetting/missing to use the equation when getting the norm or mag to normalize at the end. I think I am good. Thanks for all the quick help.
     
  8. May 4, 2009 #7

    matt grime

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    You don't just use it at the end to normalize - you use it at every occasion that you need to take an inner product.
     
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