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Math Help - Basis

  1. #1
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    Basis

    Exhibit a basis for the given space and prove that it is a basis.

    The space of 2 x 2 matrices.

    I don't understand how to this at all, can someone take me through this example step by step and kind of explain why/concepts involved.

    Thanks so much.
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  2. #2
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    Quote Originally Posted by millerst View Post
    Exhibit a basis for the given space and prove that it is a basis.

    The space of 2 x 2 matrices.

    I don't understand how to this at all, can someone take me through this example step by step and kind of explain why/concepts involved.

    Thanks so much.
    It's the 2x2 identity matrix, since it is linearly independent and spanning.
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  3. #3
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    I can't understand what Roam said at all since the words "independent" and "spanning" apply to sets of vectors, not individual vectors or matrices.

    millerst, you probably know that a basis for, say, R^4, the set of quadruples, (a, b, c, d), is {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} since we can write (a, b, c, d)= (a, 0, 0, 0)+ (0, b, 0, 0)+ (0, 0, c, 0)+ (0, 0, 0, d)= a(1, 0, 0, 0)+ b(0, 1, 0, 0)+ c(0, 0, 1, 0)+ d(0, 0, 0, 1).

    Okay, any matrix in your space can be written as
    \begin{bmatrix}a & b \\ c & d\end{bmatrix}

    You want matrices M1, M2, M3, M4 so that
    \begin{bmatrix}a & b \\ c & d\end{bmatrix}= aM1+ bM2+ cM3+ dM4
    there are obvious matrices that do that.
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  4. #4
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    The standard basis for M_{22} consists of the following four matrices:

     \left\{ \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} \right\}

    Because the space of 2x2 matrices has dimension 4, so can't be spanned by just two vectors.

    Sorry for my carelessness...
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