Editing Exterior algebra (section)

{{Short description|Algebra associated to any vector space}}
{{redirect|Wedge product|the operation on topological spaces|Wedge sum}}
{{multiple image
   | left
   | footer    = Geometric interpretation of grade ''n'' elements in a real exterior algebra for {{nowrap|1=''n'' = 0}} (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume).  The exterior product of ''n'' vectors can be visualized as any ''n''-dimensional shape (e.g. ''n''-[[Parallelepiped#Parallelotope|parallelotope]], ''n''-[[ellipsoid]]); with magnitude ([[hypervolume]]), and [[Orientation (vector space)|orientation]] defined by that of its {{nowrap|(''n'' − 1)}}-dimensional boundary and on which side the interior is.<ref name=Penrose07>{{cite book |first=R. |last=Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=978-0-679-77631-4}}</ref><ref>{{harvnb|Wheeler|Misner |Thorne|1973|p=83}}</ref>
   | width1    = 220
   | image1    = N vector positive.svg
   | caption1  = Orientation defined by an ordered set of vectors.
   | width2    = 220
   | image2    = N vector negative.svg
   | caption2  = Reversed orientation corresponds to negating the exterior product.
  }}
In mathematics, the '''exterior algebra''' or '''Grassmann algebra''' of a [[vector space]] <math>V</math> is an [[associative algebra]] that contains <math>V,</math> which has a product, called '''exterior product''' or '''wedge product''' and denoted with <math>\wedge</math>, such that <math>v\wedge v=0</math> for every vector <math>v</math> in <math>V.</math> The exterior algebra is named after [[Hermann Grassmann]],<ref>{{harvcoltxt|Grassmann|1844}} introduced these as ''extended'' algebras (cf. {{harvnb|Clifford|1878}}).</ref> and the names of the product come from the "wedge" symbol <math>\wedge</math> and the fact that the product of two elements of <math>V</math> is "outside" <math>V.</math> 

The wedge product of <math>k</math> vectors <math>v_1 \wedge v_2 \wedge \dots \wedge v_k</math> is called a ''[[blade (geometry)|blade]] of degree <math>k</math>'' or ''<math>k</math>-blade''. The wedge product was introduced originally as an algebraic construction used in [[geometry]] to study [[area]]s, [[volume]]s, and their higher-dimensional analogues: the [[magnitude (mathematics)|magnitude]] of a [[bivector|{{math|2}}-blade]] <math>v\wedge w</math> is the area of the [[parallelogram]] defined by <math>v</math> and <math>w,</math> and, more generally, the magnitude of a <math>k</math>-blade is the (hyper)volume of the [[Parallelepiped#Parallelotope|parallelotope]] defined by the constituent vectors. The [[alternating algebra|alternating property]] that <math>v\wedge v=0</math> implies a skew-symmetric property that <math>v \wedge w = -w \wedge v,</math> and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation. 

The full exterior algebra contains objects that are not themselves blades, but [[linear combination]]s of blades; a sum of blades of homogeneous degree <math>k</math> is called a [[Multivector|{{mvar|k}}-''vector'']], while a more general sum of blades of arbitrary degree is called a ''[[multivector]]''.<ref>The term ''k-vector'' is not equivalent to and should not be confused with similar terms such as ''[[4-vector]]'', which in a different context could mean an element of a 4-dimensional vector space.  A minority of authors use the term <math>k</math>-multivector instead of <math>k</math>-vector, which avoids this confusion.</ref> The [[linear span]] of the <math>k</math>-blades is called the <math>k</math>-''th exterior power'' of <math>V.</math> The exterior algebra is the [[direct sum]] of the <math>k</math>-th exterior powers of <math>V,</math> and this makes the exterior algebra a [[graded algebra]].

The exterior algebra is [[universal property|universal]] in the sense that every equation that relates elements of  <math>V</math> in the exterior algebra is also valid in every associative algebra that  contains <math>V</math>  and in which the square of every element of <math>V</math> is zero.

The definition of the exterior algebra can be extended for spaces built from vector spaces, such as [[vector field]]s and [[function (mathematics)|functions]] whose [[domain of a function|domain]] is a vector space. Moreover, the field of [[scalar (mathematics)|scalar]]s may be any field. More generally, the exterior algebra can be defined for [[module (mathematics)|modules]] over a [[commutative ring]]. In particular, the algebra of [[differential forms]] in <math>k</math> variables is an exterior algebra over the ring of the [[smooth function]]s in <math>k</math> variables.