Recall, that a matrix is called *symmetric* if $A^T = A$. Write down a basis in the space of *symmetric* $2\times 2$ matrices (there are many possible answers). How many elements are in the basis?

Solutionn: A symmetric $2\times 2$ matrix $A$ must have the form

$$

A=\begin{pmatrix} a & b \\ b & c\end{pmatrix}.

$$ Define the following matrices

$$

{\bf e}_1=\begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix},

$$ $$

{\bf e}_2=\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix},

$$ $$

{\bf e}_3=\begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix}.

$$ Then the matrix $A$ above can be uniquely expressed as

$$

A=a{\bf e}_1+b{\bf e}_2+c{\bf e}_3.

$$ Therefore ${\bf e}_1,{\bf e}_2,{\bf e}_3$ is a basis in the space of symmetric $2\times 2$ matrices.