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Math Help - Proof of a small lemma (systems of linear equations)

  1. #1
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    Proof of a small lemma (systems of linear equations)

    Suppose v (a vector) is a solution to Ax=0, and Q is an element of the reals. Show Qv is also a solution.

    I went about this by saying that Av = 0 (by the equation), and that A(Qv) = Av + Av + Av + Av + .... up to Q times, which = 0 + 0 + 0 + 0 + ... = 0
    I don't think this is a proper way of proving this fact, though...can anyone point me in the right direction?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by mistykz View Post
    Suppose v (a vector) is a solution to Ax=0, and Q is an element of the reals. Show Qv is also a solution.

    I went about this by saying that Av = 0 (by the equation), and that A(Qv) = Av + Av + Av + Av + .... up to Q times, which = 0 + 0 + 0 + 0 + ... = 0
    I don't think this is a proper way of proving this fact, though...can anyone point me in the right direction?
    \bold v is a solution to \bold A\bold x=\bold 0. We need to show that Q\bold v is also a solution, where Q\in\mathbb{R}.

    Substituting \bold x=Q\bold v into the equation, we see that \bold A(Q\bold v)=0\implies Q(\bold A\bold v)=\bold 0

    Since \bold A\bold v=\bold 0, we now see that Q(\bold 0)=\bold 0\implies \bold 0=\bold 0.

    We have verified that Q\bold v is a solution to \bold A\bold x=\bold 0

    \mathbb{Q.E.D.}

    I hope this makes sense!

    --Chris
    Last edited by Chris L T521; September 7th 2008 at 09:34 AM.
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