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Thread: 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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    $\displaystyle \begin{bmatrix}1&3&5&0\\0&1&4&-2\end{bmatrix}$

    Find your rref:

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

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

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

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

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

    Therefore, Null(A) =

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

    and

    $\displaystyle \begin{bmatrix}-6\\2\\0\\1\end{bmatrix}$
    Last edited by galactus; Nov 30th 2006 at 11:12 AM.
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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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