= Symplectic basis
{title2=$(e_1,\ldots,e_n,f_1,\ldots,f_n)$}
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= Symplectic bases
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A <symplectic basis> organizes a <nondegenerate> <alternating bilinear form> into independent pairs. Let $(V,\omega)$ be a <finite-dimensional vector space> over a <field> $K$, with $\omega$ a <nondegenerate bilinear form> satisfying $\omega(v,v)=0$ for every <vector> $v$. Its <dimension> is even, say $2n$, and a <symplectic basis> is an ordered <basis> $\mathcal B=(e_1,\ldots,e_n,f_1,\ldots,f_n)$ such that
$$
\omega(e_i,e_j)=\omega(f_i,f_j)=0,\qquad
\omega(e_i,f_j)=\delta_{ij},\qquad
\omega(f_i,e_j)=-\delta_{ij}.
$$
Here $\delta_{ij}$ is the <Kronecker delta>. Each $(e_i,f_i)$ is a <hyperbolic pair>; different pairs have zero mutual pairing. These conventions apply in every <characteristic>. In <characteristic> two, $-1=1$, but the condition $\omega(v,v)=0$ is still essential: a <symmetric bilinear form> need not be an <alternating bilinear form>. The existence theorem and the <characteristic> caveat are given in <Bilinear Forms by Keith Conrad> (Keith Conrad, Bilinear Forms, §1, Theorem 1.6; §5, Theorems 5.1 and 5.4 and Definition 5.3).
The benefit is a common coordinate language for every <symplectic vector space> of the same finite <dimension> over the same <field>. The <span> of all $e_i$, and separately of all $f_i$, is a <Lagrangian subspace>; each paired <span> is a <symplectic subspace>. In <Hamiltonian mechanics>, the corresponding pairing of position and <momentum> coordinates describes <phase space> and <canonical transformations>. On a <symplectic manifold>, a <symplectic basis> describes one <tangent space>; <Darboux coordinates> additionally describe a whole local neighborhood.<Handbook of Differential Geometry, Volume II/Symplectic Geometry>[§1.1, pp. 82–83: paired bases and Lagrangian spans; §1.5, pp. 90–91: Darboux coordinates]{ref}
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