---
tag: QA9YR4GBNB
kind: definition
name: Geometric algebra
msc: [15A66]
defines: N8H460W3E2
uses: [A38B1QFK20, 000AYPA9AJ, 0SQZMXB6GP, 0MDMZ965VE]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_algebra
---

## Definition

Let $V$ be a finite-dimensional real [[vector space|000AYPA9AJ]] with a nondegenerate symmetric [[bilinear form|0SQZMXB6GP]] $u \cdot v$.
The *geometric algebra* $\mathcal{G}(V)$ is the [[Clifford algebra|A38B1QFK20]] of $V$ for the quadratic form $Q(v) = v \cdot v$.
So it is the associative unital real algebra generated by $V$ subject to $v v = v \cdot v$ for all $v \in V$.
Its product is the *geometric product*.
For vectors $u, v \in V$, the *outer product* is $u \wedge v = \tfrac{1}{2}(uv - vu)$, and then $uv = u \cdot v + u \wedge v$.
An element of $\mathcal{G}(V)$ is a *multivector*.
A product of $k$ pairwise orthogonal vectors is a $k$-blade, and the span of the $k$-blades is the grade $k$ part $\mathcal{G}^k(V)$.

## Remarks

Polarizing $v v = v \cdot v$ gives $uv + vu = 2\, u \cdot v$, which is why $u \cdot v = \tfrac{1}{2}(uv + vu)$.
As a graded vector space, $\mathcal{G}(V) = \bigoplus_{k=0}^{n} \mathcal{G}^k(V)$ is isomorphic to the [[exterior algebra|0MDMZ965VE]] $\Lambda V$, of dimension $2^n$ for $n = \dim V$.
Example: $V = \R^2$ with the dot product and orthonormal basis $e_1, e_2$.
Then $e_1 e_2 = -e_2 e_1$, and $I = e_1 e_2$ satisfies $I^2 = e_1 e_2 e_1 e_2 = -e_1 e_1 e_2 e_2 = -1$, so $\set{a + bI}$ is a copy of $\C$.
With $R = \cos(\theta/2) - I \sin(\theta/2)$ and $\tilde R = \cos(\theta/2) + I \sin(\theta/2)$, one checks $R e_1 \tilde R = e_1 (\cos \theta + I \sin \theta) = \cos\theta \, e_1 + \sin\theta \, e_2$, a rotation by $\theta$.
Non-example: the exterior algebra $\Lambda V$ with the wedge product alone is not a geometric algebra in this sense, because there $v \wedge v = 0$ for every $v$, while $v v = v \cdot v \ne 0$ for a non-null vector.
The notation $\mathcal{G}(p, q)$ or $\mathrm{Cl}_{p,q}(\R)$ is used when the form has $p$ positive and $q$ negative squares.