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Homework Help: Special subspace of M(2*3) (R)

  1. Aug 23, 2010 #1
    [tex]W=Sp\{\left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} 1 & 0 & 1 \\ 2 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} -1 & 1 & -1 \\ -3 & -2 & -3 \end{array} \right) \}[/tex]

    I have to find subspace T, so that [tex]M_{2*3}(R)=W\oplus T[/tex]

    I solved it by finding 5 liner independent matrices (in relation to matrices in W) and made them basis for T.

    I'll appreciate any ideas.
     
  2. jcsd
  3. Aug 23, 2010 #2

    lanedance

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    not sure if I'm reading it correctly but don't you need to find the matricies orthogonal to those in W

    you will need 6 linearly independent matricies to span M_2,3
     
  4. Aug 23, 2010 #3
    Maybe I should clear myself.

    I have to find a liner space T so that, [tex]T+W=M_{2*3}[/tex] and [tex]T \cap W=0[/tex]

    I've noticed that matricies from [tex]

    W=Sp\{\left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} 1 & 0 & 1 \\ 2 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} -1 & 1 & -1 \\ -3 & -2 & -3 \end{array} \right) \}
    [/tex] are liner dependent, so I have to find 4 more independent matrices.
     
  5. Aug 23, 2010 #4

    lanedance

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    ok, so whats the issue?

    note if it helps you can wirte them as 6-vectors and do normal gram-schimdt...
     
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