EXTERIOR ALGEBRA
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(Redirected from Exterior product)
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. In linear algebra, the exterior product provides an abstract algebraic basis-independent manner for describing the determinant and the minors of a linear transformation, and is fundamentally related to ideas of rank and linear independence. The 'exterior algebra' (also known as the 'Grassmann algebra', after Hermann Grassmann[1]) of a given vector space ''V'' is the algebra 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.
Formally, the exterior algebra is a certain unital associative algebra over a field ''K'' that contains ''V'' as a subspace. It is denoted by Λ(''V'') or Λ•(''V'') and its multiplication is also known as the ''wedge product'' or the ''exterior product'' and is written as . The wedge product is an associative and bilinear operation
:.
:::.
Its essential feature is that it is ''alternating'' on ''V'':
:(1)
which implies in particular
:(2) for all , and
:(3) whenever are linearly dependent.[2]
In terms of category theory, the exterior algebra is a type of functor on vector spaces, given by a universal construction. It 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. With the additional structure of a volume form, the exterior algebra becomes a Hopf algebra whose antipode is the Hodge dual. In many cases, the exterior algebra is naturally realized as a certain subspace of the tensor algebra of ''V''.
The Cartesian plane 'R'2 is a vector space equipped with a basis consisting of a pair of unit vectors
:
Suppose that
:
are a pair of given vectors in 'R'2, written in components. There is a unique parallelogram having 'v' and 'w' as two of its sides. The ''area'' of this parallelogram is given by the standard determinant formula:
:
Consider now the exterior product of 'v' and 'w':
:
:
where the first step uses the distributive law for the wedge product, and the last uses the fact that the wedge product is alternating. Note that the coefficient in this last expression is precisely the determinant of the matrix ['v' 'w']. The fact that this may be positive or negative has the intuitive meaning that 'v' and 'w' may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the 'signed area' of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation.
The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if A('v','w') denotes the signed area of the parallelogram determined by the pair of vectors 'v' and 'w', then A must satisfy the following properties:
# A(''a'''v',''b'''w') = ''a b'' A('v','w') for any real numbers ''a'' and ''b'', since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram).
# A('v','v') = 0, since the area of the degenerate parallelogram determined by 'v' (i.e., a line segment) is zero.
# A('w','v') = -A('v','w'), since interchanging the roles of 'v' and 'w' reverses the orientation of the parallelogram.
# A('v' + ''a'''w','w') = A('v','w'), since adding a multiple of 'w' to 'v' affects neither the base nor the height of the parallelogram and consequently preserves its area.
# A('e'1, 'e'2) = 1, since the area of the unit square is one.
With the exception of the last property, the wedge product satisfies the same formal properties as the area. In a certain sense, the wedge product generalizes the final property by allowing the area of a parallelogram to be compared to that of any "standard" chosen parallelogram. In other words, the exterior product in two-dimensions is a ''basis-independent'' formulation of area.[3]
For vectors in 'R'3, the exterior algebra is closely related to the cross product and triple product. Using the standard basis {'e'1, 'e'2, 'e'3}, the wedge product of a pair of vectors
:
and
:
is
:
where {'e'1 Λ 'e'2, 'e'1 Λ 'e'3, 'e'2 Λ 'e'3} is the basis for the three-dimensional space Λ2('R'3). This imitates the usual definition of the cross product of vectors in three dimensions.
Bringing in a third vector
:,
the wedge product of three vectors is
:
where 'e'1 Λ 'e'2 Λ 'e'3 is the basis vector for the one-dimensional space Λ3('R'3). This imitates the usual definition of the triple product.
The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product 'u'×'v' can be interpreted as a vector which is perpendicular to both 'u' and 'v' and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the minors of the matrix with columns 'u' and 'v'. The triple product of 'u', 'v', and 'w' is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns 'u', 'v', and 'w'. The exterior product in three-dimensions allows for similar interpretations. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions.
The exterior algebra Λ(''V'') over a vector space ''V'' is defined as the quotient algebra of the tensor algebra by the two-sided ideal ''I'' generated by all elements of the form such that ''x'' ∈ ''V''. Symbolically,
:
The wedge product ∧ of two elements of Λ(''V'') is defined by
:
This product is anticommutative on elements of ''V'', for supposing that ''x'', ''y'' ∈ ''V'',
:
whence
:
More generally, if ''x''1, ''x''2, ..., ''x''k are elements of ''V'', and σ is a permutation of the integers [1,...,''k''], then
:
where sgn(σ) is the signature of the permutation σ.
The ''k''-th 'exterior power' of ''V'', denoted Λ''k''(''V''), is the vector subspace of Λ(''V'') spanned by elements of the form
:
If α ∈ Λ''k''(''V''), then α is said to be a ''k''-'multivector'. If, furthermore, α can be expressed as a wedge product of ''k'' elements of ''V'', then α is said to be 'decomposable'. Although decomposable multivectors span Λ''k''(''V''), not every element of Λ''k''(''V'') is decomposable. For example, in 'R'4, the following 2-multivector is not decomposable:
:
(This is in fact a symplectic form. To see this, one need only check that α ∧ α ≠0.)
If the dimension of ''V'' is ''n'' and {''e''1,...,''e''''n''} is a basis of ''V'', then the set
:
is a basis for 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 combination 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 do not 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 minors of the matrix 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 the binomial coefficient ''n'' choose ''k''. In particular, Λ''k''(''V'') = {0} for ''k'' > ''n''.
Any element of the exterior algebra can be written as a sum of multivectors. Hence, as a vector space the exterior algebra is a 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 2''n''.
The wedge product of a ''k''-multivector with a ''p''-multivector is a (''k''+''p'')-multivector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section
:
gives the exterior algebra the additional structure of a graded algebra. Symbolically,
:
Moreover, the wedge product is graded anticommutative, meaning that if α ∈ Λk(''V'') and β ∈ Λp(''V''), then
:
Let ''V'' be a vector space over the field ''K''. Informally, multiplication in Λ(''V'') is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identities ''v'' ∧ ''v'' = 0 for ''v'' ∈ ''V'' and ''v'' ∧ ''w'' = -''w'' ∧ ''v'' for ''v'', ''w'' ∈ ''V''. Formally, Λ(''V'') is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative ''K''-algebra containing ''V'' with alternating multiplication on ''V'' must contain a homomorphic image of Λ(''V''). In other words, the exterior algebra has the following universal property:[4]
Given any unital associative ''K''-algebra ''A'' and any ''K''-linear map ''j'' : ''V'' → ''A'' such that ''j''(''v'')''j''(''v'') = 0 for every ''v'' in ''V'', then there exists ''precisely one'' unital algebra homomorphism ''f'' : Λ(''V'') → ''A'' such that ''f''(''v'') = ''j''(''v'') for all ''v'' in ''V''.
To construct the most general algebra that contains ''V'' and whose multiplication is alternating on ''V'', it is natural to start with the most general algebra that contains ''V'', the tensor algebra ''T''(''V''), and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided ideal ''I'' in ''T''(''V'') generated by all elements of the form ''v''⊗''v'' for ''v'' in ''V'', and define Λ(''V'') as the quotient
:Λ(''V'') = T(''V'')/''I''
(and use Λ as the symbol for multiplication in Λ(''V'')). It is then straightforward to show that Λ(''V'') contains ''V'' and satisfies the above universal property.
As a consequence of this construction, the operation of assigning to a vector space ''V'' its exterior algebra Λ(''V'') is a functor from the category of vector spaces to the category of algebras.
Rather than defining Λ(''V'') first and then identifying the exterior powers Λ''k''(''V'') as certain subspaces, one may alternatively define the spaces Λ''k''(''V'') first and then combine them to form the algebra Λ(''V''). This approach is often used in differential geometry and is described in the next section.
Given a commutative ring ''R'' and an ''R''-module ''M'', we can define the exterior algebra Λ(''M'') just as above, as a suitable quotient of the tensor algebra 'T'(''M''). It will satisfy the analogous universal property. Many of the properties of Λ(''M'') also require that ''M'' be a projective module. Where finite-dimensionality is used, the properties further require that ''M'' be finitely generated and projective.
Given two vector spaces ''V'' and ''X'', an 'alternating operator' (or ''anti-symmetric operator'') from ''V''''k'' to ''X'' is a multilinear map
:''f'': ''V''''k'' → ''X ''
such that whenever ''v''1,...,''v''''k'' are linearly dependent vectors in ''V'', then
:''f''(''v''1,...,''v''''k'') = 0.
The most famous example is the determinant, an alternating operator from (''K''''n'')''n'' to ''K''.
The map
:''w'': ''V''''k'' → Λ''k''(''V'')
which associates to ''k'' vectors from ''V'' their wedge product, i.e. their corresponding ''k''-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on ''V''''k'': given any other alternating operator ''f'' : ''V''''k'' → ''X'', there exists a unique linear map φ: Λ''k''(''V'') → ''X'' with ''f'' = φ o ''w''. This universal property characterizes the space Λ''k''(''V'') and can serve as its definition.
The above discussion specializes to the case when ''X'' = ''K'', the base field. In this case an alternating multilinear function
:''f'' : ''V''k → ''K''
is called an 'alternating multilinear form'. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the multiplication of such a map with a scalar, is again alternating. If ''V'' has finite dimension ''n'', then this space can be identified with Λ''k''(''V''∗), where ''V''∗ denotes the dual space of ''V''. In particular, the dimension of the space of anti-symmetric maps from ''V''''k'' to ''K'' is the binomial coefficient ''n'' choose ''k''.
Under this identification, and if the base field is 'R' or 'C', the wedge product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : ''V''''k'' → ''K'' and η : ''V''''m'' → ''K'' are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their wedge product is the sum of the numbers of their variables. It is defined as follows:
:
where the alternation Alt of a multilinear map is defined to be the signed average of the values over all the permutations of its variables:
:
This definition of the wedge product is well-defined even if the fields ''K'' has finite characteristic, if
one considers an equivalent version of the above that does not use factorials or any constants:
:,
where here is the subset of ''k,m shuffles'': permutations
sending to numbers , and to numbers
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. In linear algebra, the exterior product provides an abstract algebraic basis-independent manner for describing the determinant and the minors of a linear transformation, and is fundamentally related to ideas of rank and linear independence. The 'exterior algebra' (also known as the 'Grassmann algebra', after Hermann Grassmann[1]) of a given vector space ''V'' is the algebra 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.
Formally, the exterior algebra is a certain unital associative algebra over a field ''K'' that contains ''V'' as a subspace. It is denoted by Λ(''V'') or Λ•(''V'') and its multiplication is also known as the ''wedge product'' or the ''exterior product'' and is written as . The wedge product is an associative and bilinear operation
:.
:::.
Its essential feature is that it is ''alternating'' on ''V'':
:(1)
which implies in particular
:(2) for all , and
:(3) whenever are linearly dependent.[2]
In terms of category theory, the exterior algebra is a type of functor on vector spaces, given by a universal construction. It 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. With the additional structure of a volume form, the exterior algebra becomes a Hopf algebra whose antipode is the Hodge dual. In many cases, the exterior algebra is naturally realized as a certain subspace of the tensor algebra of ''V''.
Motivating examples
Areas in the plane
The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices.
:
Suppose that
:
are a pair of given vectors in 'R'2, written in components. There is a unique parallelogram having 'v' and 'w' as two of its sides. The ''area'' of this parallelogram is given by the standard determinant formula:
:
Consider now the exterior product of 'v' and 'w':
:
:
where the first step uses the distributive law for the wedge product, and the last uses the fact that the wedge product is alternating. Note that the coefficient in this last expression is precisely the determinant of the matrix ['v' 'w']. The fact that this may be positive or negative has the intuitive meaning that 'v' and 'w' may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the 'signed area' of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation.
The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if A('v','w') denotes the signed area of the parallelogram determined by the pair of vectors 'v' and 'w', then A must satisfy the following properties:
# A(''a'''v',''b'''w') = ''a b'' A('v','w') for any real numbers ''a'' and ''b'', since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram).
# A('v','v') = 0, since the area of the degenerate parallelogram determined by 'v' (i.e., a line segment) is zero.
# A('w','v') = -A('v','w'), since interchanging the roles of 'v' and 'w' reverses the orientation of the parallelogram.
# A('v' + ''a'''w','w') = A('v','w'), since adding a multiple of 'w' to 'v' affects neither the base nor the height of the parallelogram and consequently preserves its area.
# A('e'1, 'e'2) = 1, since the area of the unit square is one.
With the exception of the last property, the wedge product satisfies the same formal properties as the area. In a certain sense, the wedge product generalizes the final property by allowing the area of a parallelogram to be compared to that of any "standard" chosen parallelogram. In other words, the exterior product in two-dimensions is a ''basis-independent'' formulation of area.[3]
Cross and triple products
For vectors in 'R'3, the exterior algebra is closely related to the cross product and triple product. Using the standard basis {'e'1, 'e'2, 'e'3}, the wedge product of a pair of vectors
:
and
:
is
:
where {'e'1 Λ 'e'2, 'e'1 Λ 'e'3, 'e'2 Λ 'e'3} is the basis for the three-dimensional space Λ2('R'3). This imitates the usual definition of the cross product of vectors in three dimensions.
Bringing in a third vector
:,
the wedge product of three vectors is
:
where 'e'1 Λ 'e'2 Λ 'e'3 is the basis vector for the one-dimensional space Λ3('R'3). This imitates the usual definition of the triple product.
The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product 'u'×'v' can be interpreted as a vector which is perpendicular to both 'u' and 'v' and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the minors of the matrix with columns 'u' and 'v'. The triple product of 'u', 'v', and 'w' is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns 'u', 'v', and 'w'. The exterior product in three-dimensions allows for similar interpretations. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions.
Formal definitions and algebraic properties
The exterior algebra Λ(''V'') over a vector space ''V'' is defined as the quotient algebra of the tensor algebra by the two-sided ideal ''I'' generated by all elements of the form such that ''x'' ∈ ''V''. Symbolically,
:
The wedge product ∧ of two elements of Λ(''V'') is defined by
:
Anticommutativity of the wedge product
This product is anticommutative on elements of ''V'', for supposing that ''x'', ''y'' ∈ ''V'',
:
whence
:
More generally, if ''x''1, ''x''2, ..., ''x''k are elements of ''V'', and σ is a permutation of the integers [1,...,''k''], then
:
where sgn(σ) is the signature of the permutation σ.
The exterior power
The ''k''-th 'exterior power' of ''V'', denoted Λ''k''(''V''), is the vector subspace of Λ(''V'') spanned by elements of the form
:
If α ∈ Λ''k''(''V''), then α is said to be a ''k''-'multivector'. If, furthermore, α can be expressed as a wedge product of ''k'' elements of ''V'', then α is said to be 'decomposable'. Although decomposable multivectors span Λ''k''(''V''), not every element of Λ''k''(''V'') is decomposable. For example, in 'R'4, the following 2-multivector is not decomposable:
:
(This is in fact a symplectic form. To see this, one need only check that α ∧ α ≠0.)
Basis and dimension
If the dimension of ''V'' is ''n'' and {''e''1,...,''e''''n''} is a basis of ''V'', then the set
:
is a basis for 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 combination 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 do not 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 minors of the matrix 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 the binomial coefficient ''n'' choose ''k''. In particular, Λ''k''(''V'') = {0} for ''k'' > ''n''.
Any element of the exterior algebra can be written as a sum of multivectors. Hence, as a vector space the exterior algebra is a 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 2''n''.
Graded structure
The wedge product of a ''k''-multivector with a ''p''-multivector is a (''k''+''p'')-multivector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section
:
gives the exterior algebra the additional structure of a graded algebra. Symbolically,
:
Moreover, the wedge product is graded anticommutative, meaning that if α ∈ Λk(''V'') and β ∈ Λp(''V''), then
:
Universal property
Let ''V'' be a vector space over the field ''K''. Informally, multiplication in Λ(''V'') is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identities ''v'' ∧ ''v'' = 0 for ''v'' ∈ ''V'' and ''v'' ∧ ''w'' = -''w'' ∧ ''v'' for ''v'', ''w'' ∈ ''V''. Formally, Λ(''V'') is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative ''K''-algebra containing ''V'' with alternating multiplication on ''V'' must contain a homomorphic image of Λ(''V''). In other words, the exterior algebra has the following universal property:[4]
Given any unital associative ''K''-algebra ''A'' and any ''K''-linear map ''j'' : ''V'' → ''A'' such that ''j''(''v'')''j''(''v'') = 0 for every ''v'' in ''V'', then there exists ''precisely one'' unital algebra homomorphism ''f'' : Λ(''V'') → ''A'' such that ''f''(''v'') = ''j''(''v'') for all ''v'' in ''V''.
Universal property of the exterior algebra
To construct the most general algebra that contains ''V'' and whose multiplication is alternating on ''V'', it is natural to start with the most general algebra that contains ''V'', the tensor algebra ''T''(''V''), and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided ideal ''I'' in ''T''(''V'') generated by all elements of the form ''v''⊗''v'' for ''v'' in ''V'', and define Λ(''V'') as the quotient
:Λ(''V'') = T(''V'')/''I''
(and use Λ as the symbol for multiplication in Λ(''V'')). It is then straightforward to show that Λ(''V'') contains ''V'' and satisfies the above universal property.
As a consequence of this construction, the operation of assigning to a vector space ''V'' its exterior algebra Λ(''V'') is a functor from the category of vector spaces to the category of algebras.
Rather than defining Λ(''V'') first and then identifying the exterior powers Λ''k''(''V'') as certain subspaces, one may alternatively define the spaces Λ''k''(''V'') first and then combine them to form the algebra Λ(''V''). This approach is often used in differential geometry and is described in the next section.
Generalizations
Given a commutative ring ''R'' and an ''R''-module ''M'', we can define the exterior algebra Λ(''M'') just as above, as a suitable quotient of the tensor algebra 'T'(''M''). It will satisfy the analogous universal property. Many of the properties of Λ(''M'') also require that ''M'' be a projective module. Where finite-dimensionality is used, the properties further require that ''M'' be finitely generated and projective.
Duality
Alternating operators
Given two vector spaces ''V'' and ''X'', an 'alternating operator' (or ''anti-symmetric operator'') from ''V''''k'' to ''X'' is a multilinear map
:''f'': ''V''''k'' → ''X ''
such that whenever ''v''1,...,''v''''k'' are linearly dependent vectors in ''V'', then
:''f''(''v''1,...,''v''''k'') = 0.
The most famous example is the determinant, an alternating operator from (''K''''n'')''n'' to ''K''.
The map
:''w'': ''V''''k'' → Λ''k''(''V'')
which associates to ''k'' vectors from ''V'' their wedge product, i.e. their corresponding ''k''-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on ''V''''k'': given any other alternating operator ''f'' : ''V''''k'' → ''X'', there exists a unique linear map φ: Λ''k''(''V'') → ''X'' with ''f'' = φ o ''w''. This universal property characterizes the space Λ''k''(''V'') and can serve as its definition.
Alternating multilinear forms
The above discussion specializes to the case when ''X'' = ''K'', the base field. In this case an alternating multilinear function
:''f'' : ''V''k → ''K''
is called an 'alternating multilinear form'. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the multiplication of such a map with a scalar, is again alternating. If ''V'' has finite dimension ''n'', then this space can be identified with Λ''k''(''V''∗), where ''V''∗ denotes the dual space of ''V''. In particular, the dimension of the space of anti-symmetric maps from ''V''''k'' to ''K'' is the binomial coefficient ''n'' choose ''k''.
Under this identification, and if the base field is 'R' or 'C', the wedge product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : ''V''''k'' → ''K'' and η : ''V''''m'' → ''K'' are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their wedge product is the sum of the numbers of their variables. It is defined as follows:
:
where the alternation Alt of a multilinear map is defined to be the signed average of the values over all the permutations of its variables:
:
This definition of the wedge product is well-defined even if the fields ''K'' has finite characteristic, if
one considers an equivalent version of the above that does not use factorials or any constants:
:,
where here is the subset of ''k,m shuffles'': permutations
sending to numbers , and to numbers
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