---
tag: 08HK38GCSM
kind: definition
name: Exterior algebra
msc: [15A75]
defines: 0MDMZ965VE
uses: [05EGHEQZAW, 0G31VJ5YS5]
sources:
  - url: https://en.wikipedia.org/wiki/Exterior_algebra
    locator: Formal definition
---

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
Let $T(V) = \bigoplus_{k \ge 0} V^{\otimes k}$ be the tensor algebra of $V$, where $V^{\otimes 0} = F$ and the product is the [[tensor product|0BF2DE2SYK]].
Let $I$ be the two-sided ideal of $T(V)$ generated by the elements $v \otimes v$ for $v \in V$.
The *exterior algebra* of $V$ is the quotient algebra $\Lambda(V) = T(V) / I$.
The product in $\Lambda(V)$ is written $\wedge$.

## Remarks

For all $u, v \in V$, $v \wedge v = 0$ and $u \wedge v = -v \wedge u$.
The image of $V^{\otimes k}$ is the $k$-th exterior power $\Lambda^k(V)$; if $\dim V = n$, then $\dim \Lambda^k(V) = \binom{n}{k}$ and $\dim \Lambda(V) = 2^n$.
If $(e_1, \dots, e_n)$ is a basis of $V$, then $\Lambda^n(V)$ is spanned by $e_1 \wedge \cdots \wedge e_n$.
For $v_j = \sum_i a_{ij} e_i$, $v_1 \wedge \cdots \wedge v_n = \det(a_{ij})\, e_1 \wedge \cdots \wedge e_n$.

## Explanations

### L1 human

The exterior algebra multiplies vectors with the wedge product, which changes sign when we swap two factors.
In $\R^3$, $u \wedge v$ describes an oriented parallelogram, and $u \wedge v \wedge w$ an oriented volume.
The top wedge product of $n$ vectors gives the determinant.