---
tag: PQ7JJY0VPH
kind: concept
name: Symplectic vector space
msc: [15A63, 53D05]
specializes: [000AYPA9AJ]
sources:
  - url: https://en.wikipedia.org/wiki/Symplectic_vector_space
same_as:
  wikidata: Q766774
  wikipedia: Symplectic vector space
---

## Summary

A symplectic vector space is a vector space with a nondegenerate [[alternating bilinear form|0DGDS5XY84]], called the symplectic form.
A finite-dimensional symplectic vector space has even dimension and a standard basis in which the form has one fixed shape.
The linear maps that keep the form make up the [[symplectic group|X12FS2JAP0]].
Symplectic vector spaces are the linear model of phase space in classical mechanics and of symplectic manifolds.

## Explanations

### L1 human

An inner product measures lengths and angles; a symplectic form measures signed areas instead.
On $\R^2$, the form $\omega((x_1, y_1), (x_2, y_2)) = x_1 y_2 - x_2 y_1$ gives the signed area of the parallelogram spanned by two vectors.
Every vector has area zero with itself, but each nonzero vector has nonzero area with some other vector.
In higher dimensions, the space splits into planes like this one, so the dimension is always even.