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Finding the Null Space

  1. Oct 12, 2011 #1
    1. The problem statement, all variables and given/known data

    Determine the null space of the following matrix:

    A = [1 1 -1 2
    2 2 -3 1
    -1 -1 0 -5]

    2. Relevant equations

    [itex] Ax=0 [/itex] where [itex] x = (x_{1}, x_{2}, x_{3}, x_{4})^{T}[/itex]

    3. The attempt at a solution

    If I put the system Ax=0 into augmented form:

    1 1 -1 2 | 0
    2 2 -3 1 | 0
    -1 -1 0 -5 | 0

    By row reduction I get the following row echelon form:

    1 1 -1 2 | 0
    0 0 1 3 | 0
    0 0 0 0 | 0

    So if

    [itex] x = (x_{1}, x_{2}, x_{3}, x_{4})^{T} = (-5t-s, s, -3t, t)^{T}[/itex]

    [itex] = t(-5, 0, -3, 1)^{T} + s(-1, 1, 0, 0)^{T}[/itex]

    [itex] = Span[ (-5, 0, -3, 1)^{T}, (-1, 1, 0, 0)^{T} ][/itex]

    My book has the answer:

    [itex] x = (-1, 1, 0, 0)^{T}, Span[ (-5, 0, -3, 1)^{T}][/itex]

    Have I gone wrong somewhere or are these answers equivalent? I can't see it if they are...
     
  2. jcsd
  3. Oct 12, 2011 #2

    Mark44

    Staff: Mentor

    The above looks fine.
    They have the same vectors you have, but their notation is screwed up. The nullspace here is two-dimensional, so it takes two vectors to span it.
     
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