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## Solution to Linear Algebra Done Wrong Exercise 1.2.3

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.