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Math Help - Null space for Matrix with similar columns

  1. #1
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    Question Null space for Matrix with similar columns

    Hi,

    I just read that the null space for the following matrix

    [1 0 1;
    5 4 9;
    2 4 6]

    is the line of all points x = c, y = c, z = -c.

    I was wondering what the null space for the matrix with three similar columns would be e.g.

    [1 1 1;
    0 0 0;
    0 0 0]

    (c, -c, 0), (-c, 2c, -c), etc are all part of the null space for this matrix but they don't form a line nor a plane.

    Any help would be much appreciated.

    Regards
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  2. #2
    MHF Contributor

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    Tejas
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    Re: Null space for Matrix with similar columns

    the matrix:

    \begin{bmatrix}1&1&1\\0&0&0\\0&0&0\end{bmatrix}

    has rank 1, so it has null space of dimension 2. it's clear that (x,y,z) is in this nullspace iff x+y+z = 0, that is if (x,y,z) is of the form: (s,t,-s-t),

    which is the plane: s(1,0,-1) + t(0,1,-1). a basis is {(1,0,-1),(0,1,-1)}.

    both of your example points lie in this plane. simply take s = c, t = 0, for the first one, and s = -c, t = 2c for the second.
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