In linear algebra, a standard symplectic basis is a basis
e
i
,
f
i
of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form
ω
, such that
ω
(
e
i
,
e
j
)
=
0
=
ω
(
f
i
,
f
j
)
,
ω
(
e
i
,
f
j
)
=
δ
i
j
. A symplectic basis of a symplectic vector space always exists; it can be constructed by a procedure similar to the Gram–Schmidt process. The existence of the basis implies in particular that the dimension of a symplectic vector space is even if it is finite.
