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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.



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