Exterior algebra
= Exterior algebra
{title2=$\Lambda^\bullet V$}
{wiki}
The <exterior algebra> of a <vector space> $V$ combines <scalars>, <vectors>, and higher-dimensional oriented elements in one <graded algebra>. In a finite-dimensional <vector space> it is
$$
\Lambda^\bullet V=\bigoplus_{k=0}^{\dim V}\Lambda^kV
$$
with multiplication given by the <exterior product>. Degree zero is the scalar <field>, degree one is $V$, and degree two is the space of <bivectors>. Multiplication adds degrees: combining two <vectors> produces a <bivector>, rather than another ordinary <vector>.<Exterior Powers by Keith Conrad>[Exterior Powers, §10: exterior algebra and its grading]{ref}
The essential rule is \b[$v\wedge v=0$]. Repeating the same direction cannot produce a nonzero oriented area. <Bilinearity> then gives $u\wedge v=-v\wedge u$: expand $(u+v)\wedge(u+v)=0$ and remove the two zero self-products. The alternating rule remains part of the definition in <characteristic two>, where sign reversal alone would not force a self-product to vanish.
If $V=\mathbb R^2$ with <standard basis> $e_1,e_2$, every element of its <exterior algebra> has the form
$$
a+be_1+ce_2+d(e_1\wedge e_2).
$$
There are four independent <coefficients>: one scalar, two vector components, and one area component. In $\mathbb R^3$, the respective degree dimensions are $1,3,3,1$, giving eight components in total. More generally, the <exterior power> basis gives total <dimension> $\sum_{k=0}^n\binom nk=2^n$.
This language is useful when direction and orientation matter together. <Determinants> measure the action on top-degree <exterior powers>; <differential forms> use exterior powers of a <dual space> to describe quantities integrated along paths, surfaces, and higher-dimensional regions. The <Grassmann algebra> viewpoint also makes the sign changes in anticommuting generators systematic. A single multiplication rule connects geometry and algebra.