Let $\omega \in \mathrm { A } ( E )$ and $\mathcal { B } = \left( b _ { 1 } , \ldots , b _ { n } \right)$ a basis of $E$.
(a) Show that there exists a unique matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } )$, whose coefficients we shall specify, such that for all $( x , y ) \in E ^ { 2 } , \omega ( x , y ) = { } ^ { t } X M Y$ where $X , Y \in \mathbb { R } ^ { n }$ are the column matrices representing respectively $x$ and $y$ in the basis $\mathcal { B }$: $$X = \left( \begin{array} { c } x _ { 1 } \\ \vdots \\ x _ { n } \end{array} \right) , \quad Y = \left( \begin{array} { c } y _ { 1 } \\ \vdots \\ y _ { n } \end{array} \right) , \quad \begin{aligned} & x = x _ { 1 } b _ { 1 } + \cdots + x _ { n } b _ { n } \\ & y = y _ { 1 } b _ { 1 } + \cdots + y _ { n } b _ { n } . \end{aligned}$$ We then denote $M = \operatorname { Mat } _ { \mathcal { B } } ( \omega )$.
(b) Show that $M$ is antisymmetric, that is, ${ } ^ { t } M = - M$.
(c) Show that the vector space $\mathrm { A } ( E )$ is of dimension 1 when $E$ is of dimension 2.
(d) Show the equivalence between the three following statements.
$\left( \mathcal { E } _ { 1 } \right) : \quad \omega$ is a symplectic form on $E$.
$\left( \mathcal { E } _ { 2 } \right) : \quad$ For all $x \in E \backslash \{ 0 \}$, there exists $y \in E$ such that $\omega ( x , y ) \neq 0$.
$\left( \mathcal { E } _ { 3 } \right) : \quad \operatorname { Mat } _ { \mathcal { B } } ( \omega )$ is invertible.