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Math Help - Find conditions on a, b, c, d

  1. #1
    MHF Contributor alexmahone's Avatar
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    Find conditions on a, b, c, d

    Find conditions on a, b, c, d \in \mathbb{Z} for {(a, b), (c, d)} to be a basis for \mathbb{Z}\times\mathbb{Z}. [Hint: Solve x(a, b) + y(c, d) = (e, f) in \mathbb{R}, and see when the x and y lie in \mathbb{Z}.]

    My solution:

    x(a, b) + y(c, d) = (e, f)
    ax + cy = e
    bx + dy = f
    x = (ed - fc)/(ad - bc)
    y = (af - be)/(ad - bc)

    Since x and y \in \mathbb{Z}, ad - bc =\pm 1 is the required condition.

    Am I right?
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  2. #2
    Super Member TheChaz's Avatar
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    Re: Find conditions on a, b, c, d

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