$M_{22}$ is the vector space of $2\times 2$ matrices. Let $S_{22}$ denote the set of all $2\times 2$ symmetric matrices. That is \begin{equation*} S_{22}=\setparts{A\in M_{22}}{\transpose{A}=A} \end{equation*}

1. Show that $S_{22}$ is a subspace of $M_{22}$.
2. Exhibit a basis for $S_{22}$ and prove that it has the required properties.
3. What is the dimension of $S_{22}$?