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Math Help - Showing that a set spans the subspace of symmetric matrices

  1. #1
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    Showing that a set spans the subspace of symmetric matrices

    Show that the set \{\begin{bmatrix}1 & 0\\ 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix},\begin{bmatrix}0 & 0\\ 0 & 1\end{bmatrix}\} spans the subspace of symmetric matrices in M_{2x2}.
    I'm not really sure how to show this. It's the part about symmetric matrices that throws me off.

    What I know:

    A symmetric matrix has the property that A = A^T. To show that the set spans, I could create a matrix and show that if there are leading ones in each row, without a pivot in the augmented column, the vectors will span the subspace:

    \begin{bmatrix}1 & 0 & 0& a\\ 0 & 1 & 0 &b\\ 0 & 1 & 0 & c\\ 0 & 0 & 1 & d\end{bmatrix} \Rightarrow \begin{bmatrix}1 & 0 & 0& 0\\ 0 & 1 & 0 &0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{bmatrix}

    Therefore, looking at the RREF of this matrix, I'll end up with a pivot in the augmented column, which doesn't lead to the right conclusion.

    As I said before, the A = A^T condition is throwing me off, since showing something like the fact that two vectors in \mathbb{R}^4 span the subspace of \mathbb{R}^4, where S is the set of all vectors whose first and last components are zero, is relatively straightforward.
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  2. #2
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    Re: Showing that a set spans the subspace of symmetric matrices

    Any symmetric matrix in $M_{2\times 2}$ can be represented by $\begin{bmatrix} a & b \\ b & c\end{bmatrix}$ where $a,b,c \in \mathbb{R}$. Consider the linear combination: $\begin{bmatrix} a & b \\ b & c\end{bmatrix} = a\begin{bmatrix} 1 & 0 \\ 0 & 0\end{bmatrix} + b \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix} + c \begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix}$
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  3. #3
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    Re: Showing that a set spans the subspace of symmetric matrices

    Quote Originally Posted by Algebrah View Post
    I'm not really sure how to show this. It's the part about symmetric matrices that throws me off.
    A symmetric matrix has the property that A = A^T. To show that the set spans, I could create a matrix and show that if there are leading ones in each row, without a pivot in the augmented column, the vectors will span the subspace:
    Is it clear to you that $\left[ {\begin{array}{*{20}{c}}a&b\\b&c\end{array}} \right]$ is a symmetric $2\times 2$ matrix for any real $a,~b,~c~?$

    Show that matrix is the linear combination of the given three.
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  4. #4
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    Re: Showing that a set spans the subspace of symmetric matrices

    Oh!

    It's stated that the subspace will be the symmetric matrices in M_{2x2}.

    So I'll start by defining an arbitrary matrix A = \begin{bmatrix}a & b \\c & d\end{bmatrix}.

    Since A = A^T,

    \begin{bmatrix}a & b \\c & d\end{bmatrix} = \begin{bmatrix}a & c \\b & d\end{bmatrix}

    Looking at the elements of these matrices, it can be seen that:

    1) a = a
    2) b = c
    3) d = d

    Therefore, the matrix can be rewritten in terms of three elements, a, b, c, where a, b, c \in \mathbb{R}:

    A = \begin{bmatrix}a & b \\b & c\end{bmatrix}

    Now it can easily be seen that the matrix A can be written as a linear combination of the three given matrices:

    a \begin{bmatrix}1 & 0 \\0 & 0\end{bmatrix} + b \begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix} + c \begin{bmatrix}0 & 0 \\0 & 1\end{bmatrix} = \begin{bmatrix}a & b \\b & c\end{bmatrix}

    As such, the given set spans the subspace of symmetric matrices in M_{2x2}.

    Thanks to you both!
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