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Math Help - Antisymmetric matrix help

  1. #1
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    Antisymmetric matrix help

    Denote the set of 2x2 antisymmetric matrices as S. By making a correspondence between elements of [M 2,2] and four-tuples (a,b,c,d), write the subset of IR^4 corresponding to S as a span. What is the dimension of this subspace?



    This question has me stumped! No idea.

    Any help is much appreciated,

    iExcavate
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  2. #2
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    Quote Originally Posted by iExcavate View Post
    Denote the set of 2x2 antisymmetric matrices as S. By making a correspondence between elements of [M 2,2] and four-tuples (a,b,c,d), write the subset of IR^4 corresponding to S as a span. What is the dimension of this subspace?



    This question has me stumped! No idea.

    Any help is much appreciated,

    iExcavate

    The matrix A=\begin{pmatrix}a&b\\c&d\end{pmatrix} is antisymmetric iff A=\begin{pmatrix}a&b\\c&d\end{pmatrix}=-\begin{pmatrix}a&c\\b&d\end{pmatrix}=-A^t , so then....what?!

    Tonio
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    Quote Originally Posted by tonio View Post
    The matrix A=\begin{pmatrix}a&b\\c&d\end{pmatrix} is antisymmetric iff A=\begin{pmatrix}a&b\\c&d\end{pmatrix}=-\begin{pmatrix}a&c\\b&d\end{pmatrix}=-A^t , so then....what?!

    Tonio
    I know these properties of an antisymmetric matrix, I am just unsure what the question is asking. Do I now make the matrix into 'vector form' by making it (a, c,b,d)?
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    Quote Originally Posted by iExcavate View Post
    I know these properties of an antisymmetric matrix, I am just unsure what the question is asking. Do I now make the matrix into 'vector form' by making it (a, c,b,d)?

    Yes. Of course, you could also "make it into vector form" as (a,b,c,d), but this is unimportant unless otherwise specified.

    Tonio
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    So for the remainder of the question:

    "Write the subset of IR^4 corresponding to S as a span. What is the dimension of this subspace?"

    What do I do here?

    (note IR^4 means R^4)
    Last edited by iExcavate; January 30th 2010 at 10:38 PM. Reason: Correction
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    Please help? Anything is appreciated.
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  7. #7
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    Quote Originally Posted by iExcavate View Post
    Please help? Anything is appreciated.

    I don't get it: you said you know the properties of antisymmetric matrices and thus you must know that, with A as in the prior post , it must be that a=d=0\,,\,\,b=-c\Longrightarrow the set of all antisymmetric 2 x 2 matrices can be identified with the set of all vectors (0,x,-x,0)\in\mathbb{R}^4 , and it's easy to see this set is actually a subspace of
    dimension 1...

    Tonio
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