Exterior algebra

Algebra associated to any vector space

In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle V,} which has a product, called exterior product or wedge product and denoted with ∧ {\displaystyle \wedge } , such that v ∧ v = 0 {\displaystyle v\wedge v=0} for every vector v {\displaystyle v} in V . {\displaystyle V.} The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol ∧ {\displaystyle \wedge } and the fact that the product of two elements of V {\displaystyle V} is "outside" V . {\displaystyle V.} The wedge product of k {\displaystyle k} vectors v 1 ∧ v 2 ∧ ⋯ ∧ v k {\displaystyle v_{1}\wedge v_{2}\wedge \dots \wedge v_{k}} is called a simple k {\displaystyle k} -vector or k {\displaystyle k} -blade.

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