Definition:Symplectic Basis

Definition
Let $\mathbb K$ be a field.

Let $(V,f)$ be a bilinear space over $\mathbb K$ of finite dimension $2n>0$.

Let $\mathcal B = (b_1, c_1, \ldots, b_n, c_n)$ be an ordered basis of $V$

Then $\mathcal B$ is symplectic :
 * $f(b_i, b_j) = f(c_i, c_j) = 0$ for all $i,j$
 * $f(b_i, c_j) = \delta_{ij}$ for all $i,j$

That is, the matrix of $f$ relative to $\mathcal B$ has the form:
 * $\begin{pmatrix}0&1\\-1&0\\&&0&1\\&&-1&0\\&&&&\ddots\\&&&&&0&1\\&&&&&-1&0\end{pmatrix}$