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Math Help - Finding a subset of S that is a basis for W.

  1. #1
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    Finding a subset of S that is a basis for W.

    Let W be the subspace of M_2x2 (R) consisting of the symmetric 2x2 matrices. The set

    S= { [<br />
\begin{pmatrix}<br />
0 & -1 \\<br />
-1 & 1 \\<br />
\end{pmatrix} \], [<br />
\begin{pmatrix}<br />
1 & 2 \\<br />
2 & 3 \\<br />
\end{pmatrix} \] [<br />
\begin{pmatrix}<br />
2 & 1 \\<br />
1 & 9 \\<br />
\end{pmatrix} \]
    [<br />
\begin{pmatrix}<br />
1 & -2 \\<br />
-2 & 4 \\<br />
\end{pmatrix} \]
    [<br />
\begin{pmatrix}<br />
-1 & 2 \\<br />
2 & -1 \\<br />
\end{pmatrix} \]}

    generates W. Find a subset of S that is a basis for W.

    (I'm not sure what all the (0)'s are next to my matrices but they aren't supposed to be there its only supposed to be the 5 matrices)
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  2. #2
    MHF Contributor

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    Quote Originally Posted by tn11631 View Post
    Let W be the subspace of M_2x2 (R) consisting of the symmetric 2x2 matrices. The set

    S= { <br />
\begin{pmatrix}<br />
0 & -1 \\<br />
-1 & 1 <br />
\end{pmatrix}, <br />
\begin{pmatrix}<br />
1 & 2 \\<br />
2 & 3 <br />
\end{pmatrix} <br />
\begin{pmatrix}<br />
2 & 1 \\<br />
1 & 9 <br />
\end{pmatrix}
    <br />
\begin{pmatrix}<br />
1 & -2 \\<br />
-2 & 4 <br />
\end{pmatrix}
    <br />
\begin{pmatrix}<br />
-1 & 2 \\<br />
2 & -1 <br />
\end{pmatrix} }

    generates W. Find a subset of S that is a basis for W.

    (I'm not sure what all the (0)'s are next to my matrices but they aren't supposed to be there its only supposed to be the 5 matrices)
    I believe it was the "\\" just before \end{pmatrix} or the unecessary [ and \] that caused the (0)s. I have removed them here.

    The simples thing to do, I think, is just try to choose matrices from the 5 given as generators that are independent.

    For example,
    <br />
\begin{pmatrix}<br />
1 & 2 \\<br />
2 & 3 <br />
\end{pmatrix}
    is not a multiple of
    <br />
\begin{pmatrix}<br />
0 & -1 \\<br />
-1 & 1 <br />
\end{pmatrix}
    so those two are independent.

    Now, can
    <br />
\begin{pmatrix}<br />
2 & 1 \\<br />
1 & 9 <br />
\end{pmatrix}
    be written as a linear combination of the first two? That is, can you find numbers, a and b, such that
    [tex]
    \begin{pmatrix}
    2 & 1 \\
    1 & 9
    \end{pmatrix}= a\begin{pmatrix}
    1 & 2 \\
    2 & 3
    \end{pmatrix} + b\begin{pmatrix}
    1 & -2 \\
    -2 & 4
    \end{pmatrix}[/MATh]?
    If so, they they are dependent and you can discard the third matrix. If not, is the fourth matrix independent of the first three?
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  3. #3
    MHF Contributor Swlabr's Avatar
    Joined
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    Posts
    1,176
    It was the \]. If you include just that here you get the (0),

    \]

    which doesn't happen when compiling a document (or, at least, it didn't happen when I tried to).

    Anyway, shouldn't it be \[ and \]?...
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