In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of vectors is used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. Also like the cross product, the exterior product is alternating, meaning that u ∧ u = 0 for all vectors u, or equivalentlyu ∧ v = -v ∧ u for all vectors u and v. In linear algebra, the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a linear transformation that is basis-independent, and is fundamentally related to ideas of rank and linear independence.

The exterior algebra of a given vector space V over a field K is the unital associative algebra Λ generated by the exterior product. It is widely used in contemporary geometry, especially differential geometry and algebraic geometry through the algebra of differential forms, as well as in multilinear algebra and related fields. In terms of category theory, the exterior algebra is a type of functor on vector spaces, given by a universal construction. The universal construction allows the exterior algebra to be defined, not just for vector spaces over a field, but also for modules over a commutative ring, and for other structures of interest. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the wedge product. This dual algebra is precisely the algebra of alternating multilinear forms on V, and the pairing between the exterior algebra and its dual is given by the interior product.

The Cartesian plane R2 is a vector space equipped with a basis consisting of a pair of unit vectors

Wikipedia, *Alternative*, http://en.wikipedia.org/wiki/Alternating_form (as of Apr 8, 2010)

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