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Math Help - Subspace Proof, finding a basis and dimension

  1. #1
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    Subspace Proof, finding a basis and dimension

    Hey guys, I have a problem that I have to do, I am having a little trouble with it.

    Consider the set of matrices
    Subspace Proof, finding a basis and dimension-equation-w.jpg

    Show that W is a subspace ofM22.

    Find a basis for W, and hence find dim(W).
    (Make sure to demonstrate that your basis is linearly independent and spans W.)

    Thanks for your help.
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  2. #2
    MHF Contributor

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    Re: Subspace Proof, finding a basis and dimension

    You have the set of all 2 by 2 matrices of the form \begin{bmatrix}a & b- d \\ b+ a & d\end{bmatrix} and you want to
    1) Show that this is a subspace of the space of all 2 by 2 matrices.
    You must show it is closed under matrix addition and scalar multiplication. The sum of two such matrices can be written
    \begin{bmatrix}u & v- w \\ v+ u & w\end{bmatrix}+\begin{bmatrix}x & y- z \\ y+ x & z\end{bmatrix}= \begin{bmatrix}u+ x & v- w+ y- z \\ v+ u+ y+ x & w+ z\end{bmatrix}
    Can that be written in the form \begin{bmatrix}a & b- d \\ b+ a & d\end{bmatrix}? What must a, b, and d be?

    The product of the number "k" and the matrix \begin{bmatrix}x & y- z\\ y+ x & z\end{bmatrix} is \begin{bmatrix}kx & k(y- z) \\ k(y+ x) & kz\end{bmatrix}.
    Can that be written in the form \begin{bmatrix}a & b- d \\ b+ a & d\end{bmatrix}? What must a, b, and d be?

    The third condition, that the subset be non-empty, can be done by showing that the 0 matrix is in the set. Take a= b= d= 0.

    2) Find a basis for this subspace.
    Well, \begin{bmatrix}a & b- d \\ b+ a & d\end{bmatrix} = \begin{bmatrix}a & 0 \\ a & 0\end{bmatrix}+ \begin{bmatrix}0 & b \\ b & 0 \end{bmatrix}+ \begin{bmatrix}0 & -d \\ 0 & d\end{bmatrix} = a\begin{bmatrix}1 & 0 \\ 1 & 0 \end{bmatrix}+ b\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}+ d\begin{bmatrix}0 & -1 \\ 0 & 1\end{bmatrix}. Does that give you any ideas?

    3) Find the dimension.
    After (2), this should be trivial.
    Last edited by HallsofIvy; April 14th 2014 at 04:42 AM.
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