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Home > Exterior algebra

for all vectors

which entails

for all vectors , and

whenever are linearly dependent.

Note that these three properties are only valid for the vectors in V, not for all elements of the algebra Λ(V).

The exterior algebra is in fact the "most general" algebra with these properties. This means that all equations that hold in the exterior algebra follow from the above properties alone. This generality of Λ(V) is formally expressed by a certain universal property, see below.

Elements of the form v₁∧v₂∧…∧v_k with v₁,…,v_k in V are called k-vectors. The subspace of Λ(V) generated by all k-vectors is known as the k-th exterior power of V and denoted by Λ^k(V). The exterior algebra can be written as the direct sum of each of the k-th powers:

The exterior product has the important property that the product of a k-vector and an l-vector is a k+l-vector. Thus the exterior algebra forms a graded algebra where the grade is given by k. These k-vectors have geometric interpretations: the 2-vector u∧v represents the oriented parallelogram with sides u and v, while the 3-vector u∧v∧w represents the oriented parallelepiped with edges u, v, and w.

Exterior powers find their main application in differential geometry, where they are used to define differential forms. As a consequence, there is a natural wedge product for differential forms. All of these concepts go back to Hermann GrassmannHermann Gunther Grassmann ( April 15, 1809 September 26, 1877) was a mathematician, physicist, linguist, scholar, and neohumanist. Hermann Grassmann was born in Stettin (by chance, on the birthday of Leonhard Euler) and died there in 1877. His father was.

1 Basis and dimension

If the dimension of V is n and {e₁,...,e_n} is a basisAbstract algebra Algebra Linear algebra In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V''. B is a mi of V, then the set

is a basis for the k-th exterior power Λ^k(V). The reason is the following: given any wedge product of the form

then every vector v_j can be written as a linear combinationIn mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalisations given at the end of of the basis vectors e_i; using the bilinearity of the wedge product, this can be expanded to a linear combination of wedge products of those basis vectors. Any wedge product in which the same basis vector appears more than once is zero; any wedge product in which the basis vectors don't appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis k-vectors can be computed as the minorIn linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. Suppose A is an m ''n matrix and k is a positive integer not larger than m and n''. A k ''k minor of A is the determinant of a k ''k matrix obtained from A by deletings of the matrixAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number that describes the vectors v_j in terms of the basis e_i.

Counting the basis elements, we see that the dimension of Λ^k(V) is n choose kIn mathematics, in particular in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number : and : (Here m denotes the factorial of m . The binomial coefficient of n and k is also written as C n. In particular, Λ^k(V) = {0} for k > n.

The exterior algebra is a graded algebra as the direct sum

(where we set Λ⁰(V) = K and Λ¹(V) = V), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2ⁿ.

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Topics: Exterior Algebra Product Wedge Anti-symmetric Lambda;(''v Basis...

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Exterior derivativeIn mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. It is important in the theory of integration on manifolds, and is the differential use Exterior algebraIn mathematics, the exterior algebra (also known as the Grassmann algebra of a given vector space V is a certain unital associative algebra which contains V as a subspace. It is denoted by Λ V or Λ• V and its multiplication, known as the wed Exterior Gateway ProtocolThe Exterior Gateway Protocol EGP is a routing protocol for the Internet originally specified in 1982 by Eric C. Rosen of Bolt, Beranek and Newman. It was first described in and formally specified in ( 1984). EGP had limited scalability and was eventually Exterior (topology)Topology General topology In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which do not meet S''. The exterior of S is denoted by ext ''S or S e''.