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