Prove that the application
$$\begin{array} { | c l l }
\mathcal { M } _ { n } ( \mathbb { R } ) & \rightarrow & \mathbb { R } \\
M & \mapsto & \operatorname { tr } ( M )
\end{array}$$
is a linear form and that
$$\forall ( A , B ) \in \left( \mathcal { M } _ { n } ( \mathbb { R } ) \right) ^ { 2 } , \quad \operatorname { tr } ( A B ) = \operatorname { tr } ( B A ) .$$