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I want to find its nullspace

  1. Feb 11, 2010 #1
    1. The problem statement, all variables and given/known data
    I have the 3x3 matrix C=(1,-1,1; 2,0,1+i; 0,1+i,-1) and I want to find its nullspace (a set of vectors that span that subspace).


    3. The attempt at a solution
    So first I have reduced the matrix to row echelon form and I got this matrix:
    (1,-1,1; 0,1,-0.5+0.5i; 0,0,0)

    How do I read off from this the nullspace of this matrix? What is a basis for this nullspace?

    By "i" I mean imaginary since this is a complex matrix.
     
  2. jcsd
  3. Feb 11, 2010 #2

    Mark44

    Staff: Mentor

    Re: Nullspace

    I ended up with a different row-reduced matrix, with no rows of zeroes.
     
  4. Feb 11, 2010 #3
    Re: Nullspace

    Are you sure? Because I used Mathematica to check the reduced row echelon form of this matrix, and it seems the rref has a row of zeros!

    Also, does the set containing (-1,0.5+0.5i,1) and (1,0,0) span the subspace?
     
    Last edited: Feb 11, 2010
  5. Feb 11, 2010 #4
    Re: Nullspace

    I tried row-reducing it again using Matlab and I still got a zero row:

    1 0 0.5 + 0.5i
    0 1 -0.5 + 0.5i
    0 0 0
     
  6. Feb 11, 2010 #5

    vela

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    Re: Nullspace

    That reduced matrix corresponds to equations

    [tex]x+(0.5+0.5 i)z = 0[/tex]
    [tex]y+(-0.5+0.5 i)z = 0[/tex]

    Solving for the other variables in terms of z, you get a solution of

    [tex]\begin{pmatrix}x\\y\\z\end{pmatrix}=z\begin{pmatrix}-0.5-0.5i\\0.5-0.5i\\1\end{pmatrix}[/tex]

    The vector multiplying the z on the RHS is a basis of the nullspace.
     
  7. Feb 12, 2010 #6

    Mark44

    Staff: Mentor

    Re: Nullspace

    I agree with your result now.
     
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