Editing Exterior algebra (section)

== Formal definition ==
The exterior algebra <math>\bigwedge (V)</math> of a vector space <math>V</math> over a [[field (mathematics)|field]] <math>K</math> is defined as the [[Quotient ring|quotient algebra]] of the [[tensor algebra]] ''T''(''V''), where

:<math>T(V)= \bigoplus_{k=0}^\infty T^kV = K\oplus V \oplus (V\otimes V) \oplus (V\otimes V\otimes V) \oplus \cdots,</math>

by the two-sided [[Ideal (ring theory)|ideal]] <math>I</math> generated by all elements of the form <math>x \otimes x</math> such that <math>x\in V</math>.<ref> This definition is a standard one. See, for instance, {{harvtxt|Mac Lane|Birkhoff|1999}}.</ref> Symbolically,
: <math>\bigwedge (V) := T(V)/I.\, </math>

The exterior product <math>\wedge</math> of two elements of <math>\bigwedge (V)</math> is defined by
: <math>\alpha\wedge\beta = \alpha\otimes\beta \pmod I.</math>