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

1. 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.

2. 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: $\begin{bmatrix} a & b \\ b & c\end{bmatrix} = a\begin{bmatrix} 1 & 0 \\ 0 & 0\end{bmatrix} + b \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix} + c \begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix}$

3. Re: Showing that a set spans the subspace of symmetric matrices

Originally Posted by Algebrah
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.

4. 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!