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Math Help - Null Space

  1. #1
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    Null Space

    Given A = [[1,3,5,0],[0,1,4,-2]] (where A is a 2x4 matrix, incase you dont know my notation):

    Find a basis for the nul(A). Then, describe the geometry of the nul(A).
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  2. #2
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    \begin{bmatrix}1&3&5&0\\0&1&4&-2\end{bmatrix}

    Find your rref:

    \begin{bmatrix}1&0&-7&6\\0&1&4&-2\end{bmatrix}

    The nullspace is the solution space to the system of homogenoeous equations Ax=0.

    x_{1}-7x_{3}+6x_{4}=0
    x_{2}+4x_{3}-2x_{4}=0

    x_{1}=7x_{3}-6x_{4}
    x_{2}=-4x_{3}+2x_{4}

    x_{1}=7s-6t
    x_{2}=-4s+2t
    x_{3}=s
    x_{4}=t

    Therefore, Null(A) =

    \begin{bmatrix}7\\-4\\1\\0\end{bmatrix}

    and

    \begin{bmatrix}-6\\2\\0\\1\end{bmatrix}
    Last edited by galactus; November 30th 2006 at 12:12 PM.
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  3. #3
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    I just don't get what the "basis" for the null is; the linear combinations? We can automatically tell that it's lin. dependent because we can't get pivots in every row, so how do you get a lin combo.
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