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Math Help - Finding two distinct vectors

  1. #1
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    [SOLVED]Finding two distinct vectors

    Hi everyone. I'm hoping someone can help me with this:

    I need to find two distinct vectors, x and y, that each have a length of 2 and when each is multiplied by the 2X2 matrix A, they yield the same result modulo 26.

    A=
    (9 5)
    (7 3)



    Any help with this is much appreciated.
    Last edited by DrNerj; September 23rd 2012 at 09:45 PM.
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  2. #2
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    Re: Finding two distinct vectors

    What kind of values are permitted for the coordinates of x and y? Real numbers? Rationals? Integers?
    By "length 2", do you mean in the ordinary sense, or in the "modulo 26" sense?
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  3. #3
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    Re: Finding two distinct vectors

    The values for x and y have to be integers modulo 26.
    And in regards to length, I mean in the ordinary sense.
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  4. #4
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    Re: Finding two distinct vectors

    There are only four vectors in \mathbb{R}^2 that have integer coordinates and length 2:

    \left\{\begin{pmatrix} 2 \\0 \end{pmatrix}, \begin{pmatrix} -2 \\0 \end{pmatrix}, \begin{pmatrix} 0 \\2 \end{pmatrix}, \begin{pmatrix} 0 \\-2 \end{pmatrix} \right\} = \left\{ \pm \begin{pmatrix} 2 \\0 \end{pmatrix}, \pm \begin{pmatrix} 0 \\2 \end{pmatrix} \right\}.

    Then matrix A operates on that set via:

    \begin{pmatrix} 9& 5 \\7& 3 \end{pmatrix} \left(\pm \begin{pmatrix} 2 \\0 \end{pmatrix} \right) = \pm \begin{pmatrix} 9& 5 \\7& 3 \end{pmatrix} \begin{pmatrix} 2 \\0 \end{pmatrix} = \pm \begin{pmatrix} 18 \\14 \end{pmatrix}, and

    \begin{pmatrix} 9& 5 \\7& 3 \end{pmatrix} \left(\pm \begin{pmatrix} 0 \\2 \end{pmatrix}\right) = \pm \begin{pmatrix} 9& 5 \\7& 3 \end{pmatrix} \begin{pmatrix} 0 \\2 \end{pmatrix} = \pm \begin{pmatrix} 10\\ 6 \end{pmatrix}.

    So A applied to the set of vectors that have integer coordinates and length 2 produces the four vectors: \left\{ \pm \begin{pmatrix} 18 \\14 \end{pmatrix}, \pm \begin{pmatrix} 10\\ 6 \end{pmatrix}\right\}.

    When looked at mod 26, those four vectors are:

    \left\{\begin{pmatrix} 18 \\14 \end{pmatrix}, \begin{pmatrix} -18 \\-14 \end{pmatrix}, \begin{pmatrix} 10\\ 6 \end{pmatrix}, \begin{pmatrix} -10\\ -6 \end{pmatrix} \right\}

    = \left\{\begin{pmatrix} 18 \\14 \end{pmatrix}, \begin{pmatrix} 8 \\12 \end{pmatrix}, \begin{pmatrix} 10\\ 6 \end{pmatrix}, \begin{pmatrix} 16\\ 20 \end{pmatrix}\right\}.

    No two of those are equal. Thus there does not exist two distinct vectors, both having integer coordinates and length two, such that the result of applying that matrix to them is equal modulo 26.
    Last edited by johnsomeone; September 23rd 2012 at 06:50 PM.
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  5. #5
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    Re: Finding two distinct vectors

    Thanks for your help!
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