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

What I know:

A symmetric matrix has the property that $\displaystyle A = A^T$. 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:

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

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 $\displaystyle A = A^T$ condition is throwing me off, since showing something like the fact that two vectors in $\displaystyle \mathbb{R}^4$ span the subspace of $\displaystyle \mathbb{R}^4$, where S is the set of all vectors whose first and last components are zero, is relatively straightforward.