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Having problems understanding nullspace

  1. Oct 12, 2009 #1
    1. The problem statement, all variables and given/known data
    If V is the subspace spanned by (1,1,1) and (2,1,0), find a matrix A that has V as its row space. Find a matrix B that has V as its nullspace


    2. Relevant equations
    Ax = 0 for a nullspace


    3. The attempt at a solution

    So straight off the bat, I think I can solve the first part. Should be simple enough right? Matrix A that has V as its row space is just the matrix
    [[1,1,1]
    [2,1,0]]

    But how do I find matrix B that has V as its nullspace?
    The only thing I can possibly think of its to do what I have done before which is to do the rref

    Which I think gives me
    [[1,0,-1]
    [0,1,2]]

    Do I run linear combination on the row space from here?


    Also, a similar question that I am having problems with is
    Find a basis for the space of a 2x3 matrices whose nullspace contains (2,1,1)

    I'm not even sure how to approach this question. My gut instinct tells me to run matrix multiplication on this one, such that

    [tex]
    \begin{pmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \end{pmatrix} \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}
    [/tex]

    This is about as far as my understanding of nullspaces carries me.
     
  2. jcsd
  3. Oct 12, 2009 #2

    HallsofIvy

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    Do the matrix multiplication
    [tex]\begin{bmatrix}a & b & c \\d & e & f\end{bmatrix}\begin{bmatrix}1 & 0 \\ 0 & 1 \\ -1 & 2\end{bmatrix}[/tex]
    and set it equal to
    [tex]\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}[/tex]
    That will give you four equations, a- c= 0, b+ 2c= 0, d- f= 0, e+ 2f= 0. That is four equations in six variables. You can solve for 4 of the variables in terms of the other 2. Notice that the problem says "find a matrix". There are an infinite number of such matrices.

    Exactly what you needed to do! (Except that the matrix on the right should have only two rows.) When you do that multiplication you get
    [tex]\begin{bmatrix} 2x_{11}+ x_{12}+ x_{13} \\ 2x_{21}+ x_{22}+ x_{23}\end{bmatrix}= \begin{bmatrix} 0 \\ 0\end{bmatrix}[/tex]
    which gives you two equation for the six variables: [itex]2x_{11}+ x_{12}+ x_{13}= 0[/itex] and [itex]2x_{21}+ x_{22}+ x_{23}= 0[/itex]. Now you can solve for 2 of those in terms of the other 4 very easily: [itex]x_{12}= -2x_{11}- x_{13}[/itex] and [itex]x_{22}= -x_{21}- x_{23}[/itex]. Put those into the original matrix:
    [tex]\begin{bmatrix} x_{11} & -2x_{11}- x_{13} & x_{13} \\ x_{21} & -2x_{21}- x_{23} & x_{23} \end{bmatrix}[/tex]
    [tex]= \begin{bmatrix} x_{11} & -2x_{11} & 0 \\ 0 & 0 & 0 \end{bmatrix}+ \begin{bmatrix}0 & -1 & 1 \\ 0 & 0 & 0\end{bmatrix}[/tex]
    [tex]+ \begin{bmatrix}0 & 0 & 0 \\ x_{21} & -2x_{21} & 0\end{bmatrix}+ \begin{bmatrix}0 & 0 & 0 \\ 0 & -x_{23} & x_{23}\end{bmatrix}[/tex]

    [tex]= x_{11}\begin{bmatrix}1 & -2 & 0 \\ 0 & 0 & 0\end{bmatrix}+ x_{13}\begin{bmatrix} 0 & 0 & 0 \\ 1 & -2 & 0\end{bmatrix}+ x_{23}\begin{bmatrix}0 & 0 & 0 \\ 0 & -1 & 1\end{bmatrix}[/tex]
    Do you get the point?
     
    Last edited: Oct 12, 2009
  4. Oct 12, 2009 #3
    So for the first part, this would mean that my solution is basically whatever combination I can get out of the equations I have. So if I make a = 1 and d = 1, then my matrix becomes

    [[1,-2,1]
    [1,-2,1]]

    So is this a possible solution to the nullspace?

    as for part two:
    this means that the basis of my nullspace would be
    [[1,2,0]
    [0,0,0]]

    and
    [[0,0,0]
    [1,-2,0]]

    and

    [[0,0,0]
    [0,-1,1]]

    Though I thought basis of a nullspace were normally written as vectors rather than matrixes? Or does this make the vectors, <0,0,0>,<0,-1,1>,<1,-2,0>
     
  5. Oct 12, 2009 #4

    HallsofIvy

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    you were asked to find a basis for "the space of a 2x3 matrices". In this case your vector space is a space of matrices. Your vectors are matrices!
     
  6. Oct 12, 2009 #5
    Oh, now I understand.
    Much thanks, you cleared a lot of this up.
     
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