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Math Help - dimensions and basis

  1. #1
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    dimensions and basis

    M =
    {( a b)
    {(-b c) :a,b,c real numbers }

    N =
    {( x 0)
    {(y 0) :x,y real numbers }

    are subspaces of M2(R).

    I need to find the dimension of M and N. how do i do this
    is the dimension the number of elements in the basis?
    if so how do i find a basis for these?
    thank you
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  2. #2
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    The basis for M is:
    \left\{ {\left[ {\begin{array}{rl}<br />
   1 & 0  \\<br />
   0 & 0  \\<br />
\end{array}} \right],\left[ {\begin{array}{rl}<br />
   0 & 1  \\<br />
   { - 1} & 0  \\<br />
\end{array}} \right],\left[ {\begin{array}{rl}<br />
   0 & 0  \\<br />
   0 & 1  \\<br />
\end{array}} \right]} \right\}.
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  3. #3
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    yeah i thought it was but i worked it out just by looking at it. if im doing a proof is there a way i can devise a basis or does it have to satisfy anything to be a basis?
    this means the dimension of M = 3?
    what is M + N?
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  4. #4
    MHF Contributor

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    The sum M+N produces M_2 (R).
    To see this consider:
    \left[ {\begin{array}{rr}<br />
   x & y  \\<br />
   w & z  \\<br />
\end{array}} \right] = x\left[ {\begin{array}{rr}<br />
   1 & 0  \\<br />
   0 & 0  \\<br />
\end{array}} \right] + y\left[ {\begin{array}{rr}<br />
   0 & 1  \\<br />
   { - 1} & 0  \\<br />
\end{array}} \right] + z\left[ {\begin{array}{rr}<br />
   0 & 0  \\<br />
   0 & 1  \\<br />
\end{array}} \right] + \left( {y + w} \right)\left[ {\begin{array}{rr}<br />
   0 & 0  \\<br />
   1 & 0  \\<br />
\end{array}} \right].
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