[[Symplectic vector space]]
# Determinant form

Over a field $\mathbb{K}$ the vector space $V=\mathbb{K}^2$ may be endowed with [[Symplectic vector space|symplectic]] structure under the so-called **determinant form** #m/def/geoalg 
$$
\begin{align*}
\omega \left(  \vtwo xy, \vtwo zw \right) = \det \begin{bmatrix}
x & z \\
y & w
\end{bmatrix}
= xw -yz
\end{align*}
$$
named for the [[Matrix determinant]].
This turns out to be a special case of the [[Standard symplectic space]].

## Properties

1. Given $A \in \End V$, $\omega(Av, Aw) = \det(A) \omega(v,w)$,
   and thus $\opn{Sp}_{2}(\mathbb{K}) = \opn{SL}_{2}(\mathbb{K})$.


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