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Math Help - Gaussian elimination

  1. #1
    Newbie
    Joined
    Jan 2010
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    Gaussian elimination

    I have a little problem with this exercise:

    Consider system of 4 equations in 4 variables over GF(2).
    <br />
p_1: x_1x_2+x_2x_3+x_2x_4+x_3x_4+x_1+x_3+1=0<br />
    <br />
p_2: x_1x_2+x_1x_3+x_1x_4+x_3x_4+x_2+x_3+1=0<br />
    <br />
p_3: x_1x_2+x_1x_3+x_2x_3+x_3x_4+x_1+x_4+1=0<br />
    <br />
p_4: x_1x_3+x_1x_4+x_2x_3+x_2x_4+1=0<br />

    Using Gaussian elimination we get:
    <br />
p'_1: x_1x_2+x_2x_3+x_2x_4+x_3x_4+x_1+x_3+1=0<br />
    <br />
 p'_2: x_1x_3+x_1x_4+x_2x_3+x_2x_4+x_1+x_2=0<br />
    <br />
 p'_3: x_1x_4+x_2x_3+x_1+x_2+x_3+x_4=0<br />
    <br />
 p'_4: x_1+x_2+1=0<br />

    How can I show this?
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  2. #2
    Junior Member
    Joined
    Oct 2009
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    70
    Use bases:
    {x_1x_2, x_1x_3, x_1x_4, x_2x_3, x_2x_4, x_3x_4, x_1, x_2, x_3, x_4, 1}.
    Then write equations in matrix form:

    |1 0 0 1 1 1 1 0 1 0 1|
    |1 1 1 0 0 1 0 1 1 0 1|
    |1 1 0 1 0 1 1 0 0 1 1| = A (sorry for that form - latex command doesn't work)
    |0 1 1 1 1 0 0 0 0 0 1|

    And now bring A into the row echelon form using Gaussian elimination.
    You get then
    p'_1, p'_2, p'_3
    But p'_4 is not the same like Yours.
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  3. #3
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    Thanks. I check this.
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  4. #4
    Junior Member
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    Oct 2009
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    You should very carefully examine papers about Mutants.
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