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Thread: Assistance with Subspaces of General Vector Spaces please

  1. #1
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    Assistance with Subspaces of General Vector Spaces please

    Hello

    Some questions I have here
    https://imgur.com/a/TRLBe

    3 b) is dimension 8 right? and subspace 6 matrices with 1's on each point and zero on others except diagonals and 2 matrices with 1 and -1 on diagonals?

    then 3c and 3d I have no idea haha


    Plox assist
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  2. #2
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    Re: Assistance with Subspaces of General Vector Spaces please

    3b) not quite

    $W_{3,3} = -(a+e)$

    so we have

    $a\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}+
    b\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}+
    c\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}+
    d\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}+
    e\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}+
    f\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}+
    g\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix}+
    h\begin{pmatrix}0&0&0\\0&0&0\\0&1&0\end{pmatrix}+
    (a+e)\begin{pmatrix}0&0&0\\0&0&0\\0&0&-1\end{pmatrix} =$



    $a\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}+
    b\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}+
    c\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}+
    d\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}+
    e\begin{pmatrix}0&0&0\\0&1&0\\0&0&-1\end{pmatrix}+
    f\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}+
    g\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix}+
    h\begin{pmatrix}0&0&0\\0&0&0\\0&1&0\end{pmatrix}$

    and yes it's of dimension 8


    3c) A little thought will show that $W = \begin{pmatrix}0 &a &b \\ -a &0 &c \\-b &-c &0 \end{pmatrix}$

    you should be able to apply the method of (3b) to this to come up with the basis and dimension

    3d) I'm not completely sure about this but I believe

    $dim(V)=9,~dim(W)=3,~codim_V(W)=9-3=6$ There are also 6 conditions on $W$

    $W_{1,1}=W_{2,2}=W_{3,3}=0$

    $W_{2,1}=-W_{1,2},~W_{3,1}=-W_{1,3},~W_{3,2}=-W_{2,3}$
    Last edited by romsek; Sep 17th 2017 at 02:06 AM.
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