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Math Help - Stuck solving set of linear equations [not solved]

  1. #1
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    Stuck solving set of linear equations [not solved]

    Hi all, first post.

    I'm stuck trying to solve this set of linear equations. I need to find the coefficients a, b and c and express them in terms of x, y, z.

    I've tried setting c=1 and finding expressions for a and b but when I check these with arbitrary numbers I just keep getting it wrong.
    Somebody has suggested reducing the 3 eigenvectors to row echelon form but I don't understand why. If possible, I would rather solve this algebraically.

    The vectors being multiplied by a, b and c are normalised eigenvectors which I am going to assume are correct!

    a*(1/sqrt2,0,-1/sqrt2) + b*(1/sqrt2,-sqrt2/2,1/2) + c*(1/2,sqrt2/2,1/2) =(x,y,z)

    Any help or advice would be appreciated!
    Thanks Although this is university work, it was set as a revison exercise so apologies if this in the wrong section!
    Last edited by freddiesel; October 3rd 2009 at 04:08 AM. Reason: possibly posted in wrong section
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  2. #2
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    I'll try to help until someone else gets to it.

    I get that you have three vectors that are unit vectors. As your final answer though you want to have something like x=?, y=?, z=? in terms of a,b and c?

    Are x,y and z just unit vectors themselves? I'm not quite following what you want to solve. It sounds like you are just stuck with the algebra, which I might be able to help with, but I'm lost conceptually to what you meant.

    Welcome to MHF by the way!
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  3. #3
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    Thanks for the fast response!
    Sorry for being unclear. (x,y,z) is defined as an arbitrary column vector.

    I need to find expressions for a,b,c. ie. a=... b=... c=... in terms of x,y,z for any vector (x,y,z).

    That is how I interpreted the question anyway, but just incase I'll write this part of the question down exactly as it is written:


    (...after having found normalised eigenvectors p, q, r of a matrix...)

    "Show that an arbitrary column vector x=(x,y,z) can be written as a linear combination of the three eigevectors, i.e.

    x=ap+bq+cr

    and find expressions for the coefficients a,b and c"


    Where p,q and r are the unit vectors in my first post.

    Hope I've explained that well enough now
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  4. #4
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    That helps a lot. One more thing though...

    Your three unit vectors are written in terms of (x,y,z) components so the three equations you get from the problem are like:

    x=ax_1 + bx_2+cx_3, where x_n is the x component from each vector. y and z are formed the same way.

    Am I understanding this right?
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  5. #5
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    Yes, that's the form I'd been trying solve them from. Thanks
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  6. #6
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    Ok so the three equations we get are:

    x= a( \frac{\sqrt{2}}{2})+b(\frac{\sqrt{2}}{2})+(\frac{1  }{2})c

    y=a(0)-b(\frac{\sqrt{2}}{2})+c(\frac{\sqrt{2}}{2})

    z= -a(\frac{\sqrt{2}}{2})+b(\frac{1}{2})+c(\frac{1}{2}  )

    Wow this is gross. RREF would definitely be quicker.

    I'll just post this and think some more...

    EDIT: I would definitely multiply everything by 2 to start.

    2x= a(\sqrt{2})+b(\sqrt{2})+c

    2y=a(0)-b(\sqrt{2})+c(\sqrt{2})

    2z= -a(\sqrt{2})+b+c
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  7. #7
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    Wow this is gross. RREF would definitely be quicker.
    Lol. I'll try the RREF method then if you think it's quicker. I can reduce it to the identity matrix but how does this help?

    I could post each row operation if that's any good?
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  8. #8
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    What about using the A^{-1}B method? A is a 3x3 matrix of coefficients plus variables a,b and c. B is a 3x1 matrix of [x,y,z]. The resulting matrix will be a 3x1 matrix and I think it should solve for a,b and c.

    Do you have to do this by hand? Try plugging this into a calculator to see if it works. I think it should.
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