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Math Help - Orthonormal Basis

  1. #1
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    Orthonormal Basis

    Here is an study problem for my final exam tomorrow:

    a) Find an orthonormal basis for R3 that includes a vector parallel to
    (0, 3, 4)Transpose.

    b) Find the representation of (3, 0, 4) with respect to your basis from a).

    c) Verify Plancherel's formula for this vector using this representation.

    I am pretty sure I can do part c), but I want to make sure I get everything correct up to that point. Any help is much appreciated. Thanks in advance!
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  2. #2
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    Quote Originally Posted by ktcyper03 View Post
    Here is an study problem for my final exam tomorrow:

    a) Find an orthonormal basis for R3 that includes a vector parallel to
    (0, 3, 4)Transpose.
    \begin{bmatrix}<br />
1 & 0 & 1\\ <br />
0 & 1 & 3\\ <br />
0 & 0 & 4<br />
\end{bmatrix}

    Now just orthonormalize it.
    Last edited by dwsmith; May 13th 2010 at 09:22 PM.
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  3. #3
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    how did you derive that matrix?
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  4. #4
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    Quote Originally Posted by ktcyper03 View Post
    how did you derive that matrix?
    You said you wanted an orthonormal basis for \mathbb{R}^3 so I first checked to see if \begin{bmatrix}<br />
0\\ <br />
3\\ <br />
4 <br />
\end{bmatrix} was lin ind to the standard basis of \mathbb{R}^2.

    I did that because you only gave me one vector so I assumed I could pick the other two at random.

    To find a vector parallel, I just visualized the vector and shifted it over.
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  5. #5
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    ok i understand a) now. how about part b) "the representation of (3, 0, 4) with respect to my basis from a)"

    ...if you try the problem, could you include the answer in your post? I figure if I know the solution I can figure out the problem eventually by trial and error. Without the solutions i just cant tell if correct or not, and dont know when to stop.

    Again, thanks for all your help.
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  6. #6
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    Quote Originally Posted by ktcyper03 View Post
    ok i understand a) now. how about part b) "the representation of (3, 0, 4) with respect to my basis from a)"

    ...if you try the problem, could you include the answer in your post? I figure if I know the solution I can figure out the problem eventually by trial and error. Without the solutions i just cant tell if correct or not, and dont know when to stop.

    Again, thanks for all your help.
    When you find the normalized basis, I will look it but you need to post some attempt.
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  7. #7
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    Quote Originally Posted by dwsmith View Post
    \begin{bmatrix}<br />
1 & 0 & 1\\ <br />
0 & 1 & 3\\ <br />
0 & 0 & 4<br />
\end{bmatrix}

    Now just normalize it.
    Wait a second: are you saying that we should use the basis \{(1,0,0); (0,1,0); (1,3,4)\}? First, I don't see a vector parallel to (0,3,4). Second, the basis is not orthogonal (the dot product between the second and third vectors is not zero).

    We need to apply the Gram-Schmidt orthogonalization procedure.
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  8. #8
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    that is the basis for R3 parallel to (0, 3, 4). It is not orthonormal yet until I orthonormalize it which I understand how to do. dwSmith, I'm not up to representations yet, and I figured I would post my questions while people are still awake, but I'm going to go back and work on it and post what I get. Again thank you so much.
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  9. #9
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    Quote Originally Posted by roninpro View Post
    Wait a second: are you saying that we should use the basis \{(1,0,0); (0,1,0); (1,3,4)\}? First, I don't see a vector parallel to (0,3,4). Second, the basis is not orthogonal (the dot product between the second and third vectors is not zero).

    We need to apply the Gram-Schmidt orthogonalization procedure.
    Yes, I just came up with 3 lin ind vectors in R^3 ktcypher03 can orthonormalize the vectors. How aren't (0,3,4)^T and (1,3,4)^T not parallel?
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  10. #10
    Senior Member roninpro's Avatar
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    Ah, okay, fair enough.

    My definition of "parallel" involves one vector being a multiple of another. Is there some other definition that I am not aware of?
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  11. #11
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    Quote Originally Posted by roninpro View Post
    Ah, okay, fair enough.

    My definition of "parallel" involves one vector being a multiple of another. Is there some other definition that I am not aware of?
    The cross product should be zero. I was thinking of parallel in the plane.

    A parallel vector would then be a multiple of the original.
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  12. #12
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    ok so i performed grahm-schmidt on the parallel basis for r3.

    my orthogonal family came out to y1=(1,0,0) y2=(0,1,0) and y3=(0,0,4)

    i then orthonormalized them and got z1=(1,0,0) z2=(0,1,0) and z3=(0,0,1)
    ... i feel like my answer isn't complicated enough...

    anyways, then i said that the representation of (3,0,4) with respect to my orthonormalized basis would be Rz = 3z1 + 4z3.


    am i on the right track?
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  13. #13
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    Quote Originally Posted by ktcyper03 View Post
    ok so i performed grahm-schmidt on the parallel basis for r3.

    my orthogonal family came out to y1=(1,0,0) y2=(0,1,0) and y3=(0,0,4)

    i then orthonormalized them and got z1=(1,0,0) z2=(0,1,0) and z3=(0,0,1)
    ... i feel like my answer isn't complicated enough...

    anyways, then i said that the representation of (3,0,4) with respect to my orthonormalized basis would be Rz = 3z1 + 4z3.


    am i on the right track?
    Ronin was correct on the parallelness (not a word but sounds right in the sentence) of the vector.
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  14. #14
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    so the basis for R3 you initially suggested doesnt work? it seems fine to me.
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  15. #15
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    Change the last vector to (0,6,8). It really doesn't matter since they are all lin. ind.
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