Symplectic vector space

Remarks

Example: on $F^{2n}$ with coordinates $(x_1, \dots, x_n, y_1, \dots, y_n)$, the standard form is $\omega(u, u') = \sum_{i=1}^n (x_i y'_i - x'_i y_i)$. It is alternating, since each term vanishes when $u = u'$. It is nondegenerate: if $u$ has $x_i \ne 0$, then $\omega(u, f_i) = x_i \ne 0$, and if $u$ has $y_i \ne 0$, then $\omega(u, e_i) = -y_i \ne 0$. Here $e_i$ and $f_i$ are the standard basis vectors for the coordinates $x_i$ and $y_i$. Non-example: the form $\omega(u, u') = x_1 y'_1 - x'_1 y_1$ on $F^3$ with coordinates $(x_1, y_1, z)$ is alternating but degenerate, because $(0, 0, 1)$ pairs to zero with every vector. Every symplectic vector space has even dimension $2n$ and a basis $e_1, \dots, e_n, f_1, \dots, f_n$ with $\omega(e_i, f_j) = \delta_{ij}$ and $\omega(e_i, e_j) = \omega(f_i, f_j) = 0$. In such a basis, the matrix of $\omega$ is $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$. In characteristic other than $2$, alternating is the same as skew-symmetric: $\omega(v, w) = -\omega(w, v)$.