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Exterior algebra - mklabgr - 09-01-2026
Exterior Algebra — Summary
Exterior algebra, also called Grassmann algebra, is an algebraic framework built from a vector space $V$ that provides a natural way to represent oriented areas, volumes, and higher-dimensional volumes. Its fundamental operation is the exterior (wedge) product, written $v\wedge w$. The defining rule is $v\wedge v=0$, which implies the antisymmetry relation $v\wedge w=-w\wedge v$. More generally, swapping two vectors in $v_1\wedge\cdots\wedge v_k$ changes its sign. An especially important consequence is that $v_1,\ldots,v_k$ are linearly dependent exactly when $v_1\wedge\cdots\wedge v_k=0$. Geometrically, $v\wedge w$ represents the oriented parallelogram generated by $v$ and $w$, while $u\wedge v\wedge w$ represents an oriented three-dimensional volume.
The algebra is divided into exterior powers $\bigwedge^k(V)$. If $\dim V=n$, then $\bigwedge^k(V)$ has basis elements of the form $e_{i_1}\wedge\cdots\wedge e_{i_k}$ with $i_1<\cdots<i_k$, and therefore
$dim⋀k(V)=(nk).\dim\bigwedge^k(V)=\binom{n}{k}$.
The complete exterior algebra is
$⋀(V)=⋀0(V)⊕⋀1(V)⊕⋯⊕⋀n(V),\bigwedge(V)=\bigwedge^0(V)\oplus\bigwedge^1(V)\oplus\cdots\oplus\bigwedge^n(V)$,
so its total dimension is $2^n$. Multiplication respects the grading: if $\alpha\in\bigwedge^k(V)$ and $\beta\in\bigwedge^p(V)$, then $\alpha\wedge\beta\in\bigwedge^{k+p}(V)$ and $\alpha\wedge\beta=(-1)^{kp}\beta\wedge\alpha$. Formally, exterior algebra can be obtained from the tensor algebra $T(V)$ by imposing the relation $v\otimes v=0$.
Exterior algebra is powerful because it unifies many familiar constructions. Determinants and matrix minors can be interpreted through exterior products: the determinant measures how a linear transformation scales the highest-dimensional oriented volume. The formalism also generalizes the cross and scalar triple products beyond three dimensions. Most importantly, exterior algebra forms the algebraic foundation of differential forms, making it fundamental in differential geometry, integration on manifolds, topology and mathematical physics. In relativity and electromagnetism, for example, the electromagnetic field can naturally be represented as a differential $2$-form $F=dA$.
Key takeaways
- The fundamental identity is $v\wedge v=0$, giving $v\wedge w=-w\wedge v$.
- $v_1\wedge\cdots\wedge v_k$ represents an oriented $k$-dimensional volume.
- $v_1,\ldots,v_k$ are linearly dependent iff $v_1\wedge\cdots\wedge v_k=0$.
- If $\dim V=n$, then $\dim\bigwedge^k(V)=\binom nk$ and $\dim\bigwedge(V)=2^n$.
- Exterior algebra provides a unified language for determinants, differential forms, geometry and physics.
Source:Wikipedia — Exterior algebra