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Math Help - Space, basis, dimension, rank of Matrices

  1. #1
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    Space, basis, dimension, rank of Matrices

    I'll start off easy.

    How do you determine the dimension and a basis of the group of all 2x2 matrices over R. That would be M_{2 \times 2}^R =                      \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd  a&\delta\end{array}\right)  |   (\alpha, \beta, \lambda, \delta)\in R\right\}
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  2. #2
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    Quote Originally Posted by jayshizwiz View Post
    I'll start off easy.

    How do you determine the dimension and a basis of the group of all 2x2 matrices over R. That would be M_{2 \times 2}^R = \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd  a&\delta\end{array}\right) | (\alpha, \beta, \lambda, \delta)\in R\right\}

    Well, as it is, or should be, well known, M_{n\times n}(\mathbb{R})\cong \mathbb{R}^{n^2} (isomorphism as vector spaces, among other things), so...

    Tonio
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  3. #3
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    Quote Originally Posted by jayshizwiz View Post
    I'll start off easy.

    How do you determine the dimension and a basis of the group of all 2x2 matrices over R. That would be M_{2 \times 2}^R =                      \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd  a&\delta\end{array}\right)  |   (\alpha, \beta, \lambda, \delta)\in R\right\}
    \begin{pmatrix}\alpha & \beta \\ \gamma & \delta\end{pmatrix} =\alpha \begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}+ \beta\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}+ \gamma\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}+ \delta\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}.

    It's that simple!

    One letter wrong!
    Last edited by HallsofIvy; June 10th 2010 at 09:34 AM.
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  4. #4
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    Wink

    [QUOTE]LaTeX Error: Syntax error

    It's that simple![QUOTE]

    I wish it was that simple! (:
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  5. #5
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    The extra 1 after the beta is a mistake, correct?
    ... \beta \left(\begin{array}{cc}0&1\\0&0\end{array}\right)
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  6. #6
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    Yes, and thanks for catching that!

    It should be
    <br />
\begin{pmatrix}\alpha & \beta \\ \gamma & \delta\end{pmatrix}<br />
=\alpha \begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}+ \beta\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}+ \gamma\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}+ \delta\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}<br />

    Now, it should be easy!
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