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Math Help - Basis and dimension of Matrices

  1. #1
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    Basis and dimension of Matrices

    Hello, first time posting on the forum. Been here a lot in my past couple of semesters with homework help. I've browsed as much as I could for help on this problem but I haven't made any advancement. Any help would be greatly appreciated.
    The problem reads

    Find a basis and the dimension of the subspace W of V = M_{2,2}, spanned by
    A = \left(\begin{array}{cc}1&-5\\-4&2\end{array}\right) B = \left(\begin{array}{cc}1&1\\-1&5\end{array}\right) C = \left(\begin{array}{cc}2&-4\\-5&7\end{array}\right) D = \left(\begin{array}{cc}1&-7\\-5&1\end{array}\right).

    TBH I barely have any idea where to start here. I understand bases and I can do them when they involve 1x1 vectors, but these are 2x2 and have no idea how to set it up. TIA

    DP
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  2. #2
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    Quote Originally Posted by illness View Post
    Hello, first time posting on the forum. Been here a lot in my past couple of semesters with homework help. I've browsed as much as I could for help on this problem but I haven't made any advancement. Any help would be greatly appreciated.
    The problem reads

    Find a basis and the dimension of the subspace W of V = M_{2,2}, spanned by
    A = \left(\begin{array}{cc}1&-5\\-4&2\end{array}\right) B = \left(\begin{array}{cc}1&1\\-1&5\end{array}\right) C = \left(\begin{array}{cc}2&-4\\-5&7\end{array}\right) D = \left(\begin{array}{cc}1&-7\\-5&1\end{array}\right).

    TBH I barely have any idea where to start here. I understand bases and I can do them when they involve 1x1 vectors, but these are 2x2 and have no idea how to set it up. TIA

    DP
    You have to find out how many, and what, matrices here are linearly independent and then those matrices span W. For example, and as a little help, note that C = A + B ==> C is lin. dependent on A,B and thus we can count it out. Check now what happens if we take A,B,D...

    Tonio
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  3. #3
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    I'm not quite sure I follow. It's just I don't know what to do with the matrices.
    To be a little clearer, as far as I understand with vectors and such, is that you set one free variable =1 and the rest 0 and thats your basis. <- taking the solution from Ax=0.

    I'm sure the process is different because unless the approach is something along the lines of c1A + c2B + c3C + c4D = 0, how do i play around with the matrices into something to work with?

    Dp
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