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Math Help - NullSpace

  1. #1
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    NullSpace

    Hi, can anyone help me with this question?
    Let S be the subspace of R4 given by the solution set of the equations
    x2 - 3 x3 = -x2 - x3 - x4 and x1 = x1 + x3 = x1 - 2 x4

    Find An example of a matrix for which S is the nullspace
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  2. #2
    Super Member girdav's Avatar
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    Re: NullSpace

    Write each equation on the form $a_1x_1+a_2x_2+a_3x_3+a_4x_4$ where $a_i$ are scalars. A possible matrix will "appear".
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  3. #3
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    Re: NullSpace

    for the second equation, doesn't the x1 just cancel out?
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  4. #4
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    Re: NullSpace

    your equations are equivalent to:

    2x_2 - 2x_3 + x_4 = 0
    x_3 + 2x_4 = 0
    x_3 = 0

    we can model this system of equations with the matrix equation:

    \begin{bmatrix}0&2&-2&1\\0&0&1&2\\0&0&1&0 \end{bmatrix} \begin{bmatrix}x_1\\x_2\\x_3\\x_4 \end{bmatrix} = \begin{bmatrix}0\\0\\0 \end{bmatrix}

    the fact that x1 "cancels out" is indeed relevant, it shows we are completely free to choose ANY value for it.
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  5. #5
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    Re: NullSpace

    Are you sure those are the right equations? x_1= x_1+ x_3 immediately gives x_3= 0. And x_1= x_1- 2x_4 immediately gives x_4= 0. So immediately, vectors in the solution space are of the form <x_1, x_2, 0, 0>. And your other equation, x_2- 3x_3= -x_2- x_3- x_4 becomes x_2= -x_2 which is equivalent to 2x_2= 0 or x_2= 0.
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  6. #6
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    Re: NullSpace

    it seems perfectly conceivable that A (the matrix in question) could have a 1-dimensional nullspace.
    Last edited by Deveno; March 16th 2012 at 11:02 AM.
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