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Vector Space

  1. Jun 21, 2005 #1
    Let M2 be the vector space of 2 x 2 matrices.How to find a basis for the subspace of M2 consisting of symmetric matrices.
    The problem it creates for me is that i ca guess the solution but i don't have any symstematic procedure in mind... :cry:

    Pls help
  2. jcsd
  3. Jun 21, 2005 #2


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    For Mn, you take the n matrices that are all zeroes except have a single 1 on the diagonal, plus the n(n-1)/2 matrices that have zeroes everywhere except a 1 in the i-j position and a 1 in the j-i position, where i and j are unequal. I don't think you can get any more "systematic" than this.
  4. Jun 21, 2005 #3


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    better maybe just take your guess and try to prove it is independent and spans.

    or here ios an idea: try to write down amap from some standard vector space R^t to the symmetric amtrices, in such a way that your maop is linear and an isomorphism. then it trakes a basis of the standard space to a basis of those matrices.

    i.e. map say (1,0,0) to a symmetric 2by2 matrix, and (0,1,0) to another one and (0,0,1) to another one.

    i.e. try mapping "upper triangular" matrices isomorphically to symmetricm ones.
    Last edited: Jun 21, 2005
  5. Jun 24, 2005 #4


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    Any 2 by 2 symmetric matrix must be of the form [tex]\begin{pmatrix}a & b \\ b & c\end{pmatrix}[/tex] for some numbers a, b, c.
    Taking a= 1, b= c= 0 gives [tex]\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}[/tex].
    Taking a= 0, b= 1, c= 0 gives [tex]\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}[/tex].
    Taking a= b= 0, c= 1 gives [tex]\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}[/tex].

    Those matrices form a basis for the 3 dimensional space.

    In other words, write the general matrix with constants a, b, etc. and take each succesively equal to 1, the others 0.
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