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Math Help - Matrix/basis question

  1. #1
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    Matrix/basis question

    I can see that this is true but I am having trouble presenting a convincing argument.
    Let V = M_{n\times n} (\mathbb{R}). Show that V has a basis with the property that each matrix in the basis is either symmetric or skew symmetric.

    The second part of the question however I have no idea where to start:
    Show also that V has a basis with the property that each matrix in the basis is an invertible matrix.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by worc3247 View Post
    I can see that this is true but I am having trouble presenting a convincing argument.
    Let V = M_{n\times n} (\mathbb{R}). Show that V has a basis with the property that each matrix in the basis is either symmetric or skew symmetric.
    If:

    F_1=\left\{{A\in{V}:A^t=A}\right\},\quad F_2=\left\{{A\in{V}:A^t=-A}\right\}

    then,

    V=F_1\oplus F_2

    so, the union of a basis of F_1 with a basis of F_2 is a basis of V.

    Fernando Revilla
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  3. #3
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    Quote Originally Posted by worc3247 View Post
    Let V = M_{n\times n} (\mathbb{R}).

    ...

    I have no idea where to start:
    Show also that V has a basis with the property that each matrix in the basis is an invertible matrix.
    Start with a basis for V in which the first element in the basis is the identity matrix I. You can add a multiple of I to each other element in the basis so as to make that element invertible, and the resulting set of matrices will still be a basis.
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