{"history":[],"article":{"id":"1c8a913c-38f8-445e-b376-627353df1ae0","title":"Exterior algebra","slug":"exterior-algebra","content":":::summary\n:::\n\n# \nIn mathematics, the **exterior algebra** or **Grassmann algebra** of a [vector space](https://bliptext.com/articles/vector-space) V is an [associative algebra](https://bliptext.com/articles/associative-algebra) that contains V, which has a product, called **exterior product** or **wedge product** and denoted with \\wedge, such that v\\wedge v=0 for every vector v in V. The **exterior algebra** is named after [Hermann Grassmann](https://bliptext.com/articles/hermann-grassmann), and the names of the product come from the \"wedge\" symbol \\wedge and the fact that the product of two elements of V is \"outside\" V.\nThe wedge product of k vectors is called a blade of degree k or k-blade.\nThe wedge product was introduced originally as an algebraic construction used in [geometry](https://bliptext.com/articles/geometry) to study [areas](https://bliptext.com/articles/area), [volumes](https://bliptext.com/articles/volume), and their higher-dimensional analogues: The [magnitude](https://bliptext.com/articles/magnitude-mathematics) of a [[bivector|\n2\n-blade]] v\\wedge w is the area of the [parallelogram](https://bliptext.com/articles/parallelogram) defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the [parallelotope](https://bliptext.com/articles/parallelepiped) defined by the constituent vectors.\nThe [alternating property](https://bliptext.com/articles/alternating-algebra) that v\\wedge v=0 implies a skew-symmetric property that and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.\nThe full exterior algebra contains objects that are not themselves blades, but [linear combinations](https://bliptext.com/articles/linear-combination) of blades; a sum of blades of homogeneous degree k is called a k-vector, while a more general sum of blades of arbitrary degree is called a [multivector](https://bliptext.com/articles/multivector).\nThe [linear span](https://bliptext.com/articles/linear-span) of the k-blades is called the k-th exterior power of V. The exterior algebra is the [direct sum](https://bliptext.com/articles/direct-sum) of the k-th exterior powers of V, and this makes the exterior algebra a [graded algebra](https://bliptext.com/articles/graded-algebra).\nThe exterior algebra is [universal](https://bliptext.com/articles/universal-property) in the sense that every equation that relates elements of V in the exterior algebra is also valid in every associative algebra that contains V and in which the square of every element of V is zero.\nThe definition of the exterior algebra can be extended for spaces built from vector spaces, such as [vector fields](https://bliptext.com/articles/vector-field) and [functions](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) is a vector space.\nMoreover, the field of [scalars](https://bliptext.com/articles/scalar-mathematics) may be any field (however for fields of [characteristic](https://bliptext.com/articles/characteristic-of-a-field) two, the above condition v\\wedge v=0 must be replaced with which is equivalent in other [characteristic](https://bliptext.com/articles/characteristic-of-a-field)s).\nMore generally, the exterior algebra can be defined for [modules](https://bliptext.com/articles/module-mathematics) over a [commutative ring](https://bliptext.com/articles/commutative-ring).\nIn particular, the algebra of [differential forms](https://bliptext.com/articles/differential-forms) in k variables is an exterior algebra over the ring of the [smooth functions](https://bliptext.com/articles/smooth-function) in k variables.\n\n# Motivating examples\n\n## Areas in the plane\nThe two-dimensional [Euclidean vector space](https://bliptext.com/articles/euclidean-vector-space) is a [real](https://bliptext.com/articles/real-number) vector space equipped with a [basis](https://bliptext.com/articles/basis-of-a-vector-space) consisting of a pair of orthogonal [unit vectors](https://bliptext.com/articles/unit-vector)\nSuppose that\nare a pair of given vectors in \\mathbf{R}^2, written in components.\nThere is a unique parallelogram having \\mathbf{v} and \\mathbf{w} as two of its sides.\nThe area of this parallelogram is given by the standard [determinant](https://bliptext.com/articles/determinant) formula:\nConsider now the exterior product of \\mathbf{v} and \\mathbf{w}:\nwhere the first step uses the distributive law for the [exterior product](https://bliptext.com/articles/), and the last uses the fact that the [exterior product](https://bliptext.com/articles/) is an [alternating map](https://bliptext.com/articles/alternating-map), and in particular (The fact that the [exterior product](https://bliptext.com/articles/) is an [alternating map](https://bliptext.com/articles/alternating-map) also forces ) Note that the coefficient in this last expression is precisely the determinant of the matrix [v w].\nThe fact that this may be positi**v**e or negati**v**e has the intuiti**v**e meaning that **v** and **w** may be oriented in a counterclock**w**ise or clock**w**ise sense as the **v**ertices of the parallelogram they define.\nSuch an area is called the [signed area](https://bliptext.com/articles/signed-area) of the parallelogram: the [absolute value](https://bliptext.com/articles/absolute-value) of the [signed area](https://bliptext.com/articles/signed-area) is the ordinary area, and the sign determines its orientation.\nThe fact that this coefficient is the signed area is not an accident.\nIn 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.\nIn detail, if A(**v**, **w**) denotes the signed area of the parallelogram of **w**hich the pair of **v**ectors **v** and **w** form t**w**o adjacent sides, then A must satisfy the follo**w**ing properties:\nWith the exception of the last property, the exterior product of two vectors satisfies the same properties as the area.\nIn a c****e****rtain s****e****ns****e****, th****e**** ****e****xt****e****rior product g****e****n****e****raliz****e****s th****e**** final prop****e****rty by allowing th****e**** ar****e****a of a parall****e****logram to b****e**** compar****e****d to that of any chos****e****n parall****e****logram in a parall****e****l plan****e**** (h****e****r****e****, th****e**** on****e**** with sid****e****s ****e****1 and ****e****2).\nIn other words, the exterior product provides a basis-independent formulation of area.\n# [The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices. | upload.wikimedia.org/wikipedia/commons/a/ad/Area///parallellogram///as///determinant.svg]\n# [The cross product ( blue vector) in relation to the exterior product ( light blue parallelogram). The length of the cross product is to the length of the parallel unit vector ( red ) as the size of the exterior product is to the size of the reference parallelogram ( light red ). | upload.wikimedia.org/wikipedia/commons/d/d8/Exterior///calc///cross///product.svg]\n\n## Cross and triple products\nFor vectors in **R**3, the exterior algebra is closely related to the [cross product](https://bliptext.com/articles/cross-product) and [triple product](https://bliptext.com/articles/triple-product).\nUsing the standard basis, the exterior product of a pair of vectors\nand\nis\nwhere is the basis for the three-dimensional space ⋀2(**R**3).\nThe coefficients above are the same as those in the usual definition of the [cross product](https://bliptext.com/articles/cross-product) of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a [bivector](https://bliptext.com/articles/bivector).\nBringing in a third vector\nth******e****** ******e******xt******e******rior product of thr******e************e****** v******e******ctors is wh******e******r******e****** ******e******1 ∧ ******e******2 ∧ ******e******3 is th******e****** basis v******e******ctor for th******e****** on******e******-dim******e******nsional spac******e****** ⋀3(**R**3).\nThe scalar coefficient is the [triple product](https://bliptext.com/articles/triple-product) of the three vectors.\nThe cross product and triple product in three dimensions each admit both geometric and algebraic interpretations.\nThe cross prod**u**ct **u** × **v** can be interpreted as a **v**ector which is perpendic**u**lar to both **u** and **v** and whose magnit**u**de is eq**u**al to the area of the parallelogram determined by the two **v**ectors.\nIt can also be interpreted as the **v**ector consisting of the [minors](https://bliptext.com/articles/minor-mathematics) of the matrix with col**u**mns **u** and **v**.\nThe triple prod**u**ct of **u**, **v**, and **w** is geometrically a (signed) **v**ol**u**me.\nAlgebraically, it is the determinant of the matrix **w**ith col**u**mns **u**, **v**, and **w**.\nThe exterior product in three dimensions allows for similar interpretations.\nIn fact, in the presence of a positively oriented [orthonormal basis](https://bliptext.com/articles/orthonormal-basis), the exterior product generalizes these notions to higher dimensions.\n# [Basis Decomposition of a 2-vector | upload.wikimedia.org/wikipedia/commons/8/85/2-vector///decomposition.png]\n\n# Formal definition\nThe exterior algebra of a vector space V over a [field](https://bliptext.com/articles/field-mathematics) K is defined as the [quotient algebra](https://bliptext.com/articles/quotient-ring) of the [tensor algebra](https://bliptext.com/articles/tensor-algebra) T(V), where\nby the two-sided [ideal](https://bliptext.com/articles/ideal-ring-theory) I generated by all elements of the form x \\otimes x such that x\\in V. Symbolically,\nThe exterior product \\wedge of two elements of is defined by\n\n# Algebraic properties\n\n## Alternating product\nThe exterior product is by construction alternating on elements of V, which means that for all x \\in V, by the above construction.\nIt follows that the product is also [anticommutative](https://bliptext.com/articles/anticommutative) on elements of V, for supposing that x, y \\in V, hence\nMore generally, if \\sigma is a [permutation](https://bliptext.com/articles/permutation-group) of the integers [1, \\dots, k], and x_1, x_2, ..., x_k are elements of V, it follows that\nwhere is the [signature of the permutation](https://bliptext.com/articles/signature-of-a-permutation) \\sigma.\nIn particular, if x_i = x_j for some i \\ne j, then the following generalization of the alternating property also holds:\nTogether with the distributive property of the exterior product, one further generalization is that a necessary and sufficient condition for to be a linearly dependent set of vectors is that\n\n## Exterior power\nThe\nk\nth **exterior power** of V, denoted {\\textstyle\\bigwedge}^{\\!k}(V), is the [vector subspace](https://bliptext.com/articles/vector-subspace) of {\\textstyle\\bigwedge}(V) [spanned](https://bliptext.com/articles/linear-span) by elements of the form\nIf \\alpha \\in {\\textstyle\\bigwedge}^{\\!k}(V), then \\alpha is said to be a '''[[p-vector|\nk\n-vector]]'''.\nIf, furthermore, \\alpha can be expressed as an exterior product of k elements of V, then \\alpha is said to be **decomposable** (or **simple**, by some authors; or a **blade**, by others).\nAlthough decomposable k-vectors span {\\textstyle\\bigwedge}^{\\!k}(V), not every element of is decomposable.\nFor example, given \\mathbf{R}^4 with a basis \\{ e_1, e_2, e_3, e_4 \\}, the following 2-vector is not decomposable:\n\n### Basis and dimension\nIf the [dimension](https://bliptext.com/articles/dimension-linear-algebra) of V is n and is a [basis](https://bliptext.com/articles/basis-linear-algebra) for V, then the set\nis a basis for {\\textstyle\\bigwedge}^{\\!k}(V).\nThe reason is the following: given any exterior product of the form\nevery vector v_j can be written as a [linear combination](https://bliptext.com/articles/linear-combination) of the basis vectors e_i; using the bilinearity of the exterior product, this can be expanded to a [linear combination](https://bliptext.com/articles/linear-combination) of exterior products of those basis vectors.\nAny exterior product in which the same basis vector appears more than once is zero; any exterior 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.\nIn general, the resulting coefficients of the basis k-vectors can be computed as the [minors](https://bliptext.com/articles/minor-linear-algebra) of the [matrix](https://bliptext.com/articles/matrix-mathematics) that describes the vectors v_j in terms of the basis e_i.\nBy counting the basis elements, the dimension of is equal to a [binomial coefficient](https://bliptext.com/articles/binomial-coefficient):\nwhere n is the dimension of the vectors, and k is the number of vectors in the product.\nThe binomial coefficient produces the correct result, even for exceptional cases; in particular, for k > n.\nAny element of the exterior algebra can be written as a sum of [[p-vector|\nk\n-vector]]s.\nHence, as a vector space the exterior algebra is a [direct sum](https://bliptext.com/articles/direct-sum-of-modules)\n(where, by convention, {\\textstyle\\bigwedge}^{\\!0}(V) = K, the [field](https://bliptext.com/articles/field-mathematics) underlying V, and {\\textstyle\\bigwedge}^{\\!1}(V) = V), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2^n.\n\n### Rank of a k-vector\nIf \\alpha \\in {\\textstyle\\bigwedge}^{\\!k}(V), then it is possible to express \\alpha as a linear combination of decomposable [[p-vector|\nk\n-vector]]s:\nwhere each is decomposable, say\nThe **rank** of the\nk\n-vector \\alpha is the minimal number of decomposable\nk\n-vectors in such an expansion of \\alpha.\nThis is similar to the notion of [tensor rank](https://bliptext.com/articles/tensor-rank).\nRank is particularly important in the study of 2-vectors.\nThe rank of a 2-vector \\alpha can be identified with half the [rank of the matrix](https://bliptext.com/articles/rank-of-a-matrix) of coefficients of \\alpha in a basis.\nThus if e_i is a basis for V, then \\alpha can be expressed uniquely as where (the matrix of coefficients is [skew-symmetric](https://bliptext.com/articles/skew-symmetric-matrix)).\nThe rank of the matrix a_{ij} is therefore even, and is twice the rank of the form \\alpha.\nIn characteristic 0, the 2-vector \\alpha has rank p if and only if\n\n## Graded structure\nThe exterior product of a\nk\n-vector with a\np\n-vector is a (k + p)-vector, once again invoking bilinearity.\nAs a consequence, the direct sum decomposition of the preceding section\ngives the exterior algebra the additional structure of a [graded algebra](https://bliptext.com/articles/graded-algebra), that is\nMoreover, if\nK\nis the base field, we have\nThe exterior product is graded anticommutative, meaning that if and \\beta \\in {\\textstyle\\bigwedge}^{\\!p}(V), then\nIn addition to studying the graded structure on the exterior algebra, studies additional graded structures on exterior algebras, such as those on the exterior algebra of a [graded module](https://bliptext.com/articles/graded-module) (a module that already carries its own gradation).\n\n## Universal property\nLet\nV\nbe a vector space over the field\nK\n. Informally, multiplication in is performed by manipulating symbols and imposing a [distributive law](https://bliptext.com/articles/distributive-law), an [associative law](https://bliptext.com/articles/associative-law), and using the identity for\nv ∈ V\n. Formally, is the \"most general\" algebra in which these rules hold for the multiplication, in the sense that any unital associative\nK\n-algebra containing\nV\nwith alternating multiplication on\nV\nmust contain a homomorphic image of {\\textstyle\\bigwedge}(V).\nIn other words, the exterior algebra has the following [universal property](https://bliptext.com/articles/universal-property):\nGiven any unital associative\nK\n-algebra\nA\nand any\nK\n-[linear map](https://bliptext.com/articles/linear-map) j : V \\to A such that for every\nv\nin\nV\n, then there exists precisely one unital [algebra homomorphism](https://bliptext.com/articles/algebra-homomorphism) such that\nj(v) = f(i(v))\nfor all\nv\nin\nV\n(here\ni\nis the natural inclusion of\nV\nin {\\textstyle\\bigwedge}(V), see above).\nTo construct the most general algebra that contains\nV\nand whose multiplication is alternating on\nV\n, it is natural to start with the most general associative algebra that contains\nV\n, the [tensor algebra](https://bliptext.com/articles/tensor-algebra)\nT(V)\n, and then enforce the alternating property by taking a suitable [quotient](https://bliptext.com/articles/quotient-ring).\nWe thus take the two-sided [ideal](https://bliptext.com/articles/ideal-ring-theory)\nI\nin\nT(V)\ngenerated by all elements of the form\nv ⊗ v\nfor\nv\nin\nV\n, and define as the quotient\n(and use\n∧\nas the symbol for multiplication in {\\textstyle\\bigwedge}(V)). It is then straightforward to show that contains\nV\nand satisfies the above universal property.\nAs a consequence of this construction, the operation of assigning to a vector space\nV\nits exterior algebra is a [functor](https://bliptext.com/articles/functor) from the [category](https://bliptext.com/articles/category-mathematics) of vector spaces to the [category](https://bliptext.com/articles/category-mathematics) of algebras.\nRather than defining first and then identifying the exterior powers as certain subspaces, one may alternatively define the spaces first and then combine them to form the algebra {\\textstyle\\bigwedge}(V).\nThis approach is often used in differential geometry and is described in the next section.\n# [Universal property of the exterior algebra | upload.wikimedia.org/wikipedia/commons/6/6b/ExteriorAlgebra-01.png]\n\n## Generalizations\nGiven a [commutative ring](https://bliptext.com/articles/commutative-ring) R and an R-[module](https://bliptext.com/articles/module-mathematics) M, we can define the exterior algebra just as above, as a suitable quotient of the tensor algebra \\mathrm{T}(M).\nIt will satisfy the analogous universal property.\nMany of the properties of also require that M be a [projective module](https://bliptext.com/articles/projective-module).\nWhere finite dimensionality is used, the properties further require that M be [finitely generated](https://bliptext.com/articles/finitely-generated-module) and projective.\nGeneralizations to the most common situations can be found in.\nExterior algebras of [vector bundles](https://bliptext.com/articles/vector-bundle) are frequently considered in geometry and topology.\nThere are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the [Serre–Swan theorem](https://bliptext.com/articles/serre-swan-theorem).\nMore general exterior algebras can be defined for [sheaves](https://bliptext.com/articles/sheaf-mathematics) of modules.\n\n# Alternating tensor algebra\nFor a field of characteristic not 2, the exterior algebra of a vector space V over K can be canonically identified with the vector subspace of that consists of [antisymmetric tensors](https://bliptext.com/articles/antisymmetric-tensor).\nFor characteristic 0 (or higher than \\dim V), the vector space of k-linear antisymmetric tensors is transversal to the ideal I, hence, a good choice to represent the quotient.\nBut for nonzero characteristic, the vector space of K-linear antisymmetric tensors could be not transversal to the ideal (actually, for k \\geq \\operatorname{char} K, the vector space of K-linear antisymmetric tensors is contained in I); nevertheless, transversal or not, a product can be defined on this space such that the resulting algebra is isomorphic to the exterior algebra: in the first case the natural choice for the product is just the quotient product (using the available projection), in the second case, this product must be slightly modified as given below (along Arnold setting), but such that the algebra stays isomorphic with the exterior algebra, i.e. the quotient of by the ideal I generated by elements of the form x \\otimes x. Of course, for characteristic 0 (or higher than the dimension of the vector space), one or the other definition of the product could be used, as the two algebras are isomorphic (see V. I. Arnold or Kobayashi-Nomizu).\nLet be the space of homogeneous tensors of degree r. This is spanned by decomposable tensors\nThe **antisymmetrization** (or sometimes the **skew-symmetrization**) of a decomposable tensor is defined by and, when r! \\neq 0 (for nonzero characteristic field r! might be 0):\nwhere the sum is taken over the [symmetric group](https://bliptext.com/articles/symmetric-group) of permutations on the symbols \\{ 1, \\dots, r \\}.\nThis extends by linearity and homogeneity to an operation, also denoted by \\mathcal{A} and \\rm{Alt}, on the full tensor algebra \\mathrm{T}(V).\nNote that\nSuch that, when defined, is the projection for the exterior (quotient) algebra onto the r-homogeneous alternating tensor subspace.\nOn the other hand, the image is always the **alternating tensor graded subspace** (not yet an algebra, as product is not yet defined), denoted A(V).\nThis is a vector subspace of \\mathrm{T}(V), and it inherits the structure of a graded vector space from that on \\mathrm{T}(V).\nMoreover, the kernel of is precisely I^{(r)}, the homogeneous subset of the ideal I, or the kernel of \\mathcal{A} is I. When is defined, A(V) carries an associative graded product defined by (the same as the wedge product)\nAssuming K has characteristic 0, as A(V) is a supplement of I in \\mathrm{T}(V), with the above given product, there is a canonical isomorphism\nWhen the characteristic of the field is nonzero, \\mathcal{A} will do what \\rm{Alt} did before, but the product cannot be defined as above.\nIn such a case, isomorphism still holds, in spite of A(V) not being a supplement of the ideal I, but then, the product should be modified as given below ( product, Arnold setting).\nFinally, we always get A(V) isomorphic with {\\textstyle\\bigwedge}(V), but the product could (or should) be chosen in two ways (or only one).\nActually, the product could be chosen in many ways, rescaling it on homogeneous spaces as for an arbitrary sequence c(r) in the field, as long as the division makes sense (this is such that the redefined product is also associative, i.e. defines an algebra on A(V)).\nAlso note, the interior product definition should be changed accordingly, in order to keep its skew derivation property.\n\n## Index notation\nSuppose that V has finite dimension n, and that a basis e1, ..., en of V is given.\nThen any alternating tensor t ∈ Ar(V) ⊂ Tr(V) can be written in [index notation](https://bliptext.com/articles/index-notation) with the [Einstein summation convention](https://bliptext.com/articles/einstein-summation-convention) as where ti1⋅⋅⋅ir is [completely antisymmetric](https://bliptext.com/articles/antisymmetric-tensor) in its indices.\nThe exterior product of two alternating tensors t and s of ranks r and p is given by\nThe components of this tensor are precisely the skew part of the components of the tensor product s ⊗ t, denoted by square brackets on the indices:\nThe [interior product](https://bliptext.com/articles/) may also be described in index notation as follows.\nLet be an antisymmetric tensor of rank r. Then, for α ∈ V∗, \\iota_\\alpha t is an alternating tensor of rank r - 1, given by where n is the dimension of V.\n\n# Duality\n\n## Alternating operators\nGiven two vector spaces V and X and a natural number k, an **alternating operator** from Vk to X is a [multilinear map](https://bliptext.com/articles/multilinear-map)\nsuch that whenever v1, ..., vk are [linearly dependent](https://bliptext.com/articles/linearly-dependent) vectors in V, then\nThe map\nwhich associates to k vectors from V their exterior product, i.e. their corresponding k-vector, is also alternating.\nIn fact, this map is the \"most general\" alternating operator defined on V^k; given any other alternating operator there exists a unique [linear map](https://bliptext.com/articles/linear-map) with This [universal property](https://bliptext.com/articles/universal-property) characterizes the space of alternating operators on V^k and can serve as its definition.\n\n## Alternating multilinear forms\n[[File:N-form.svg|thumb|upright=0.6|Geometric interpretation for the **exterior product** of\nn\n[1-forms](https://bliptext.com/articles/1-form) (\nε\n,\nη\n,\nω\n) to obtain an\nn\n-form (\"mesh\" of [coordinate surfaces](https://bliptext.com/articles/coordinate-surface), here planes), for\nn = 1, 2, 3\n. The \"circulations\" show [orientation](https://bliptext.com/articles/orientation-vector-space).\n]]\nThe above discussion specializes to the case when X = K, the base field.\nIn this case an alternating multilinear function\nis called an **alternating multilinear form**.\nThe set of all [alternating](https://bliptext.com/articles/alternating-map) [multilinear forms](https://bliptext.com/articles/multilinear-form) is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again [alternating](https://bliptext.com/articles/alternating-map).\nBy the universal property of the exterior power, the space of alternating forms of degree k on V is [naturally](https://bliptext.com/articles/natural-transformation) isomorphic with the [dual vector space](https://bliptext.com/articles/dual-vector-space) \\bigl({\\textstyle\\bigwedge}^{\\!k}(V)\\bigr)^*.\nIf V is finite-dimensional, then the latter is to {\\textstyle\\bigwedge}^{\\!k}\\left(V^*\\right).\nIn particular, if V is n-dimensional, the dimension of the space of alternating maps from V^k to K is the [binomial coefficient](https://bliptext.com/articles/binomial-coefficient) \\textstyle\\binom{n}{k}.\nUnder such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones.\nSuppose ω : Vk → K and η : Vm → K are two anti-symmetric maps.\nAs in the case of [tensor products](https://bliptext.com/articles/tensor-product) of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables.\nDepending on the choice of identification of elements of exterior power with multilinear forms, the exterior product is defined as or as where, if the characteristic of the base field K is 0, the alternation Alt of a multilinear map is defined to be the average of the sign-adjusted values over all the [permutations](https://bliptext.com/articles/permutation) of its variables:\nWhen the [field](https://bliptext.com/articles/field-mathematics) K has [finite characteristic](https://bliptext.com/articles/characteristic-of-a-field), an equivalent version of the second expression without any factorials or any constants is well-defined:\nwhere here Shk,m ⊂ Sk+m is the subset of [[(p,q) shuffle|\n(k, m)\nshuffles]]: [permutations](https://bliptext.com/articles/permutation) σ of the set such that σ(1) < σ(2) < ⋯ < σ(k), and σ(k + 1) < σ(k + 2) < ... < σ(k + m).\nAs this might look very specific and fine tuned, an equivalent raw version is to sum in the above formula over permutations in left cosets of Sk+m / (Sk × Sm).\n\n## Interior product\nSuppose that V is finite-dimensional.\nIf V^* denotes the [dual space](https://bliptext.com/articles/dual-space) to the vector space V, then for each \\alpha \\in V^*, it is possible to define an [antiderivation](https://bliptext.com/articles/derivation-abstract-algebra) on the algebra {\\textstyle\\bigwedge}(V),\nThis derivation is called the **interior product** with \\alpha, or sometimes the **insertion operator**, or **contraction** by \\alpha.\nSuppose that w \\in {\\textstyle\\bigwedge}^{\\!k}(V).\nThen w is a multilinear mapping of V^* to K, so it is defined by its values on the\nk\n-fold [Cartesian product](https://bliptext.com/articles/cartesian-product) V^* \\times V^* \\times \\dots \\times V^*.\nIf u1, u2, ..., uk−1 are k - 1 elements of V^*, then define\nAdditionally, let whenever f is a pure scalar (i.e., belonging to {\\textstyle\\bigwedge}^{\\!0}(V)).\n\n### Axiomatic characterization and properties\nThe interior product satisfies the following properties:\nThese three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.\nFurther properties of the interior product include:\n\n## Hodge duality\nSuppose that V has finite dimension n. Then the interior product induces a canonical isomorphism of vector spaces\nby the recursive definition\nIn the geometrical setting, a non-zero element of the top exterior power (which is a one-dimensional vector space) is sometimes called a **[volume form](https://bliptext.com/articles/volume-form)** (or **orientation form**, although this term may sometimes lead to ambiguity).\nThe name orientation form comes from the fact that a choice of preferred top element determines an orientation of the whole exterior algebra, since it is tantamount to fixing an ordered basis of the vector space.\nRelative to the preferred volume form \\sigma, the isomorphism is given explicitly by\nIf, in addition to a volume form, the vector space V is equipped with an [inner product](https://bliptext.com/articles/inner-product) identifying V with V^*, then the resulting isomorphism is called the **Hodge star operator**, which maps an element to its **Hodge dual**:\nThe composition of \\star with itself maps and is always a scalar multiple of the identity map.\nIn most applications, the volume form is compatible with the inner product in the sense that it is an exterior product of an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) of V. In this case,\nwhere id is the identity mapping, and the inner product has [metric signature](https://bliptext.com/articles/metric-signature) (p, q) — p pluses and q minuses.\n\n## Inner product\nFor V a finite-dimensional space, an [inner product](https://bliptext.com/articles/inner-product) (or a [pseudo-Euclidean](https://bliptext.com/articles/pseudo-euclidean-space) [inner product](https://bliptext.com/articles/inner-product)) on V defines an isomorphism of V with V^*, and so also an isomorphism of with \\bigl({\\textstyle\\bigwedge}^{\\!k} V\\bigr)^*.\nThe pairing between these two spaces also takes the form of an inner product.\nOn decomposable k-vectors,\nthe determinant of the matrix of inner products.\nIn the special case vi = wi, the inner product is the square norm of the k-vector, given by the determinant of the [Gramian matrix](https://bliptext.com/articles/gramian-matrix) (⟨vi, vj⟩).\nThis is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on If ei, i = 1, 2, ..., n, form an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) of V, then the vectors of the form\nconstitute an orthonormal basis for {\\textstyle\\bigwedge}^{\\!k}(V), a statement equivalent to the [Cauchy–Binet formula](https://bliptext.com/articles/cauchy-binet-formula).\nWith respect to the inner product, exterior multiplication and the interior product are mutually adjoint.\nSpecifically, for \\mathbf{v}\\in {\\textstyle\\bigwedge}^{\\!k-1}(V), \\mathbf{w} \\in {\\textstyle\\bigwedge}^{\\!k}(V), and x \\in V, where x♭ ∈ V∗ is the [musical isomorphism](https://bliptext.com/articles/musical-isomorphism), the linear functional defined by for all y \\in V. This property completely characterizes the inner product on the exterior algebra.\nIndeed, more generally for \\mathbf{v}\\in {\\textstyle\\bigwedge}^{\\!k-l}(V), \\mathbf{w}\\in {\\textstyle\\bigwedge}^{\\!k}(V), and \\mathbf{x}\\in {\\textstyle\\bigwedge}^{\\!l}(V), iteration of the above adjoint properties gives\nwhere now is the dual l-vector defined by for all \\mathbf{y} \\in {\\textstyle\\bigwedge}^{\\!l}(V).\n\n## Bialgebra structure\nThere is a correspondence between the graded dual of the graded algebra and alternating multilinear forms on V. The exterior algebra (as well as the [symmetric algebra](https://bliptext.com/articles/symmetric-algebra)) inherits a bialgebra structure, and, indeed, a [Hopf algebra](https://bliptext.com/articles/hopf-algebra) structure, from the [tensor algebra](https://bliptext.com/articles/tensor-algebra).\nSee the article on [tensor algebras](https://bliptext.com/articles/tensor-algebra) for a detailed treatment of the topic.\nThe exterior product of multilinear forms defined above is dual to a [coproduct](https://bliptext.com/articles/coproduct) defined on {\\textstyle\\bigwedge}(V), giving the structure of a [coalgebra](https://bliptext.com/articles/coalgebra).\nThe **coproduct** is a linear function \\Delta : {\\textstyle\\bigwedge}(V) \\to {\\textstyle\\bigwedge}(V) \\otimes {\\textstyle\\bigwedge}(V), which is given by on elements v \\in V. The symbol 1 stands for the unit element of the field K. Recall that K \\simeq {\\textstyle\\bigwedge}^{\\!0}(V) \\subseteq {\\textstyle\\bigwedge}(V), so that the above really does lie in {\\textstyle\\bigwedge}(V) \\otimes {\\textstyle\\bigwedge}(V).\nThis definition of the coproduct is lifted to the full space by (linear) homomorphism.\nThe correct form of this homomorphism is not what one might naively write, but has to be the one carefully defined in the [coalgebra](https://bliptext.com/articles/coalgebra) article.\nIn this case, one obtains\nExpanding this out in detail, one obtains the following expression on decomposable elements:\nwhere the second summation is taken over all [[(p,q) shuffle|\n(p, k−p)\n-shuffles]].\nBy convention, one takes that Sh(k,0) and Sh(0,k) equals {id: {1, ..., k} → {1, ..., k}}.\nIt is also convenient to take the pure wedge products and\nto equal 1 for p = 0 and p = k, respectively (the empty product in ).\nThe shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements x_k is preserved in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.\nObserve that the coproduct preserves the grading of the algebra.\nExtending to the full space one has\nThe tensor symbol ⊗ used in this section should be understood with some caution: it is not the same tensor symbol as the one being used in the definition of the alternating product.\nIntuitively, it is perhaps easiest to think it as just another, but different, tensor product: it is still (bi-)linear, as tensor products should be, but it is the product that is appropriate for the definition of a bialgebra, that is, for creating the object {\\textstyle\\bigwedge}(V) \\otimes {\\textstyle\\bigwedge}(V).\nAny lingering doubt can be shaken by pondering the equalities (1 ⊗ v) ∧ (1 ⊗ w) = 1 ⊗ (v ∧ w) and (v ⊗ 1) ∧ (1 ⊗ w) = v ⊗ w, which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols.\nThis distinction is developed in greater detail in the article on [tensor algebras](https://bliptext.com/articles/tensor-algebra).\nHere, there is much less of a problem, in that the alternating product \\wedge clearly corresponds to multiplication in the exterior algebra, leaving the symbol \\otimes free for use in the definition of the bialgebra.\nIn practice, this presents no particular problem, as long as one avoids the fatal trap of replacing alternating sums of \\otimes by the wedge symbol, with one exception.\nOne can construct an alternating product from \\otimes, with the understanding that it works in a different space.\nImmediately below, an example is given: the alternating product for the dual space can be given in terms of the coproduct.\nThe construction of the bialgebra here parallels the construction in the [tensor algebra](https://bliptext.com/articles/tensor-algebra) article almost exactly, except for the need to correctly track the alternating signs for the exterior algebra.\nIn terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:\nwhere the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely, α ∧ β = ε ∘ (α ⊗ β) ∘ Δ, where \\varepsilon is the counit, as defined presently).\nThe **counit** is the homomorphism that returns the 0-graded component of its argument.\nThe coproduct and counit, along with the exterior product, define the structure of a [bialgebra](https://bliptext.com/articles/bialgebra) on the exterior algebra.\nWith an **antipode** defined on homogeneous elements by S(x) = (-1)^{\\binom{\\text{deg}\\, x\\, + 1}{2} }x, the exterior algebra is furthermore a [Hopf algebra](https://bliptext.com/articles/hopf-algebra).\n\n# Functoriality\nSuppose that V and W are a pair of vector spaces and f : V \\to W is a [linear map](https://bliptext.com/articles/linear-map).\nThen, by the universal property, there exists a unique homomorphism of graded algebras\nsuch that\nIn particular, preserves homogeneous degree.\nThe\nk\n-graded components of are given on decomposable elements by\nLet\nThe components of the transformation {\\textstyle\\bigwedge}^{\\!k}(f) relative to a basis of V and W is the matrix of k \\times k minors of f. In particular, if V = W and V is of finite dimension n, then is a mapping of a one-dimensional vector space to itself, and is therefore given by a scalar: the [determinant](https://bliptext.com/articles/determinant) of f.\n\n## Exactness\nIf is a [short exact sequence](https://bliptext.com/articles/short-exact-sequence) of vector spaces, then\nis an exact sequence of graded vector spaces, as is\n\n## Direct sums\nIn particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:\nThis is a graded isomorphism; i.e.,\nIn greater generality, for a short exact sequence of vector spaces there is a natural [filtration](https://bliptext.com/articles/filtration-mathematics)\nwhere F^p for p \\geq 1 is spanned by elements of the form for u_i \\in U and v_i \\in V. The corresponding quotients admit a natural isomorphism\nIn particular, if U is 1-dimensional then\nis exact, and if W is 1-dimensional then\nis exact.\n\n# Applications\n\n## Oriented volume in affine space\nThe natural setting for (oriented) k-dimensional volume and exterior algebra is [affine space](https://bliptext.com/articles/affine-space).\nThis is also the intimate connection between exterior algebra and [differential forms](https://bliptext.com/articles/differential-forms), as to integrate we need a 'differential' object to measure infinitesimal volume.\nIf \\mathbb{A} is an affine space over the vector space V, and a ([simplex](https://bliptext.com/articles/simplex)) collection of ordered k+1 points, we can define its oriented k-dimensional volume as the exterior product of vectors (using concatenation PQ to mean the [displacement vector](https://bliptext.com/articles/displacement-vector) from point P to Q); if the order of the points is changed, the oriented volume changes by a sign, according to the parity of the permutation.\nIn n-dimensional space, the volume of any n-dimensional simplex is a scalar multiple of any other.\nThe sum of the (k-1)-dimensional oriented areas of the boundary simplexes of a k-dimensional simplex is zero, as for the sum of vectors around a triangle or the oriented triangles bounding the tetrahedron in the previous section.\nThe vector space structure on generalises addition of vectors in V: we have and similarly a\nk\n-blade is linear in each factor.\n\n## Linear algebra\nIn applications to [linear algebra](https://bliptext.com/articles/linear-algebra), the exterior product provides an abstract algebraic manner for describing the [determinant](https://bliptext.com/articles/determinant) and the [minors](https://bliptext.com/articles/minor-[matrix](https://bliptext.com/articles/matrix-mathematics)) of a [matrix](https://bliptext.com/articles/matrix-mathematics).\nFor instance, it is well known that the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix (with a sign to track orientation).\nThis suggests that the determinant can be defined in terms of the exterior product of the column vectors.\nLikewise, the\nk × k\nminors of a matrix can be defined by looking at the exterior products of column vectors chosen\nk\nat a time.\nThese ideas can be extended not just to matrices but to [linear transformations](https://bliptext.com/articles/linear-transformation) as well: the determinant of a linear transformation is the factor by which it scales the oriented volume of any given reference parallelotope.\nSo the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power.\nThe action of a transformation on the lesser exterior powers gives a [basis](https://bliptext.com/articles/basis-of-a-vector-space)-independent way to talk about the minors of the transformation.\n\n## Physics\nIn physics, many quantities are naturally represented by alternating operators.\nFor example, if the motion of a charged particle is described by velocity and acceleration vectors in four-dimensional spacetime, then normalization of the velocity vector requires that the electromagnetic force must be an alternating operator on the velocity.\nIts six degrees of freedom are identified with the electric and magnetic fields.\n\n## Electromagnetic field\nIn [Einstein's theories of relativity](https://bliptext.com/articles/theory-of-relativity), the [electromagnetic field](https://bliptext.com/articles/electromagnetic-field) is generally given as a [differential 2-form](https://bliptext.com/articles/differential-form) F = dA in [4-space](https://bliptext.com/articles/4-space) or as the equivalent [alternating tensor field](https://bliptext.com/articles/antisymmetric-tensor) the [electromagnetic tensor](https://bliptext.com/articles/electromagnetic-tensor).\nThen or the equivalent Bianchi identity\nNone of this requires a metric.\nAdding the [Lorentz metric](https://bliptext.com/articles/lorentz-metric) and an [orientation](https://bliptext.com/articles/orientability) provides the [Hodge star operator](https://bliptext.com/articles/hodge-star-operator) \\star and thus makes it possible to define or the equivalent tensor [divergence](https://bliptext.com/articles/divergence) where\n\n## Linear geometry\nThe decomposable\nk\n-vectors have geometric interpretations: the bivector u \\wedge v represents the plane spanned by the vectors, \"weighted\" with a number, given by the area of the oriented [parallelogram](https://bliptext.com/articles/parallelogram) with sides u and v. Analogously, the 3-vector represents the spanned 3-space weighted by the volume of the oriented [parallelepiped](https://bliptext.com/articles/parallelepiped) with edges u, v, and w.\n\n## Projective geometry\nDecomposable\nk\n-vectors in correspond to weighted\nk\n-dimensional [linear subspaces](https://bliptext.com/articles/linear-subspace) of V. In particular, the [Grassmannian](https://bliptext.com/articles/grassmannian) of\nk\n-dimensional subspaces of V, denoted \\operatorname{Gr}_k(V), can be naturally identified with an [algebraic subvariety](https://bliptext.com/articles/algebraic-variety) of the [projective space](https://bliptext.com/articles/projective-space).\nThis is called the [Plücker embedding](https://bliptext.com/articles/pl-cker-embedding), and the image of the embedding can be characterized by the [Plücker relations](https://bliptext.com/articles/pl-cker-relations).\n\n## Differential geometry\nThe exterior algebra has notable applications in [differential geometry](https://bliptext.com/articles/differential-geometry), where it is used to define [differential forms](https://bliptext.com/articles/differential-form).\nDifferential forms are mathematical objects that evaluate the length of vectors, areas of parallelograms, and volumes of [higher-dimensional bodies](https://bliptext.com/articles/parallelepiped), so they can be [integrated](https://bliptext.com/articles/integral) over curves, surfaces and higher dimensional [manifolds](https://bliptext.com/articles/manifold) in a way that generalizes the [line integrals](https://bliptext.com/articles/line-integral) and [surface integrals](https://bliptext.com/articles/surface-integral) from calculus.\nA [differential form](https://bliptext.com/articles/differential-form) at a point of a [differentiable manifold](https://bliptext.com/articles/differentiable-manifold) is an alternating multilinear form on the [tangent space](https://bliptext.com/articles/tangent-space) at the point.\nEquivalently, a differential form of degree\nk\nis a [linear functional](https://bliptext.com/articles/linear-functional) on the\nk\nth exterior power of the tangent space.\nAs a consequence, the exterior product of multilinear forms defines a natural exterior product for differential forms.\nDifferential forms play a major role in diverse areas of differential geometry.\nAn [alternate approach](https://bliptext.com/articles/differential-mathematics) defines differential forms in terms of [germs of functions](https://bliptext.com/articles/germ-mathematics).\nIn particular, the [exterior derivative](https://bliptext.com/articles/exterior-derivative) gives the exterior algebra of differential forms on a manifold the structure of a [differential graded algebra](https://bliptext.com/articles/differential-graded-algebra).\nThe exterior derivative commutes with [pullback](https://bliptext.com/articles/pullback-differential-geometry) along smooth mappings between manifolds, and it is therefore a [natural](https://bliptext.com/articles/natural-transformation) [differential operator](https://bliptext.com/articles/differential-operator).\nThe exterior algebra of differential forms, equipped with the exterior derivative, is a [cochain complex](https://bliptext.com/articles/cochain-complex) whose cohomology is called the [de Rham cohomology](https://bliptext.com/articles/de-rham-cohomology) of the underlying manifold and plays a vital role in the [algebraic topology](https://bliptext.com/articles/algebraic-topology) of differentiable manifolds.\n\n## Representation theory\nIn [representation theory](https://bliptext.com/articles/representation-theory), the exterior algebra is one of the two fundamental [Schur functors](https://bliptext.com/articles/schur-functor) on the category of vector spaces, the other being the [symmetric algebra](https://bliptext.com/articles/symmetric-algebra).\nTogether, these constructions are used to generate the [irreducible representations](https://bliptext.com/articles/irreducible-representation) of the [general linear group](https://bliptext.com/articles/general-linear-group) (see [Fundamental representation](https://bliptext.com/articles/fundamental-representation)).\n\n## Superspace\nThe exterior algebra over the complex numbers is the archetypal example of a [superalgebra](https://bliptext.com/articles/superalgebra), which plays a fundamental role in physical theories pertaining to [fermions](https://bliptext.com/articles/fermion) and [supersymmetry](https://bliptext.com/articles/supersymmetry).\nA single element of the exterior algebra is called a **supernumber** or [Grassmann number](https://bliptext.com/articles/grassmann-number).\nThe exterior algebra itself is then just a one-dimensional [superspace](https://bliptext.com/articles/superspace): it is just the set of all of the points in the exterior algebra.\nThe topology on this space is essentially the [weak topology](https://bliptext.com/articles/weak-topology), the [open sets](https://bliptext.com/articles/open-sets) being the [cylinder sets](https://bliptext.com/articles/cylinder-set).\nAn\nn\n-dimensional superspace is just the n-fold product of exterior algebras.\n\n## Lie algebra homology\nLet L be a Lie algebra over a field K, then it is possible to define the structure of a [chain complex](https://bliptext.com/articles/chain-complex) on the exterior algebra of L. This is a K-linear mapping\ndefined on decomposable elements by\nThe [Jacobi identity](https://bliptext.com/articles/jacobi-identity) holds if and only if \\partial\\partial = 0, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra L to be a Lie algebra.\nMoreover, in that case is a [chain complex](https://bliptext.com/articles/chain-complex) with boundary operator \\partial.\nThe [homology](https://bliptext.com/articles/homology-theory) associated to this complex is the Lie algebra [homology](https://bliptext.com/articles/homology-theory).\n\n## Homological algebra\nThe exterior algebra is the main ingredient in the construction of the [Koszul complex](https://bliptext.com/articles/koszul-complex), a fundamental object in [homological algebra](https://bliptext.com/articles/homological-algebra).\n\n# History\nThe exterior algebra was first introduced by [Hermann Grassmann](https://bliptext.com/articles/hermann-grassmann) in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension.\nThis referred more generally to an algebraic (or axiomatic) theory of extended quantities and was one of the early precursors to the modern notion of a [vector space](https://bliptext.com/articles/vector-space).\n[Saint-Venant](https://bliptext.com/articles/adh-mar-jean-claude-barr-de-saint-venant) also published similar ideas of exterior calculus for which he claimed priority over Grassmann.\nThe algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors.\nIt was thus a calculus, much like the [propositional calculus](https://bliptext.com/articles/propositional-calculus), except focused exclusively on the task of formal reasoning in geometrical terms.\nIn particular, this new development allowed for an axiomatic characterization of dimension, a property that had previously only been examined from the coordinate point of view.\nThe import of this new theory of vectors and [multivectors](https://bliptext.com/articles/multivector) was lost to mid-19th-century mathematicians,\nuntil being thoroughly vetted by [Giuseppe Peano](https://bliptext.com/articles/giuseppe-peano) in 1888.\nPeano's work also remained somewhat obscure until the turn of the century, when the subject was unified by members of the French geometry school (notably [Henri Poincaré](https://bliptext.com/articles/henri-poincar), [Élie Cartan](https://bliptext.com/articles/lie-cartan), and [Gaston Darboux](https://bliptext.com/articles/gaston-darboux)) who applied Grassmann's ideas to the calculus of [differential forms](https://bliptext.com/articles/differential-form).\nA short while later, [Alfred North Whitehead](https://bliptext.com/articles/alfred-north-whitehead), borrowing from the ideas of Peano and Grassmann, introduced his [universal algebra](https://bliptext.com/articles/universal-algebra).\nThis then paved the way for the 20th-century developments of [abstract algebra](https://bliptext.com/articles/abstract-algebra) by placing the axiomatic notion of an algebraic system on a firm logical footing.\n\n# Mathematical references\n\n## Historical references\n\n## Other references and further reading","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:45:15.566Z","updated_at":"2025-01-25T18:45:15.566Z","search_vector":"'algebra':2 'exterior':1","search_content":" \n \n\n# \nIn mathematics, the **exterior algebra** or **Grassmann algebra** of a [vector space](https://bliptext.com/articles/vector-space) V is an [associative algebra](https://bliptext.com/articles/associative-algebra) that contains V, which has a product, called **exterior product** or **wedge product** and denoted with \\wedge, such that v\\wedge v=0 for every vector v in V. The **exterior algebra** is named after [Hermann Grassmann](https://bliptext.com/articles/hermann-grassmann), and the names of the product come from the \"wedge\" symbol \\wedge and the fact that the product of two elements of V is \"outside\" V.\nThe wedge product of k vectors is called a blade of degree k or k-blade.\nThe wedge product was introduced originally as an algebraic construction used in [geometry](https://bliptext.com/articles/geometry) to study [areas](https://bliptext.com/articles/area), [volumes](https://bliptext.com/articles/volume), and their higher-dimensional analogues: The [magnitude](https://bliptext.com/articles/magnitude-mathematics) of a [[bivector|\n2\n-blade]] v\\wedge w is the area of the [parallelogram](https://bliptext.com/articles/parallelogram) defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the [parallelotope](https://bliptext.com/articles/parallelepiped) defined by the constituent vectors.\nThe [alternating property](https://bliptext.com/articles/alternating-algebra) that v\\wedge v=0 implies a skew-symmetric property that and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.\nThe full exterior algebra contains objects that are not themselves blades, but [linear combinations](https://bliptext.com/articles/linear-combination) of blades; a sum of blades of homogeneous degree k is called a k-vector, while a more general sum of blades of arbitrary degree is called a [multivector](https://bliptext.com/articles/multivector).\nThe [linear span](https://bliptext.com/articles/linear-span) of the k-blades is called the k-th exterior power of V. The exterior algebra is the [direct sum](https://bliptext.com/articles/direct-sum) of the k-th exterior powers of V, and this makes the exterior algebra a [graded algebra](https://bliptext.com/articles/graded-algebra).\nThe exterior algebra is [universal](https://bliptext.com/articles/universal-property) in the sense that every equation that relates elements of V in the exterior algebra is also valid in every associative algebra that contains V and in which the square of every element of V is zero.\nThe definition of the exterior algebra can be extended for spaces built from vector spaces, such as [vector fields](https://bliptext.com/articles/vector-field) and [functions](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) is a vector space.\nMoreover, the field of [scalars](https://bliptext.com/articles/scalar-mathematics) may be any field (however for fields of [characteristic](https://bliptext.com/articles/characteristic-of-a-field) two, the above condition v\\wedge v=0 must be replaced with which is equivalent in other [characteristic](https://bliptext.com/articles/characteristic-of-a-field)s).\nMore generally, the exterior algebra can be defined for [modules](https://bliptext.com/articles/module-mathematics) over a [commutative ring](https://bliptext.com/articles/commutative-ring).\nIn particular, the algebra of [differential forms](https://bliptext.com/articles/differential-forms) in k variables is an exterior algebra over the ring of the [smooth functions](https://bliptext.com/articles/smooth-function) in k variables.\n\n# Motivating examples\n\n## Areas in the plane\nThe two-dimensional [Euclidean vector space](https://bliptext.com/articles/euclidean-vector-space) is a [real](https://bliptext.com/articles/real-number) vector space equipped with a [basis](https://bliptext.com/articles/basis-of-a-vector-space) consisting of a pair of orthogonal [unit vectors](https://bliptext.com/articles/unit-vector)\nSuppose that\nare a pair of given vectors in \\mathbf{R}^2, written in components.\nThere is a unique parallelogram having \\mathbf{v} and \\mathbf{w} as two of its sides.\nThe area of this parallelogram is given by the standard [determinant](https://bliptext.com/articles/determinant) formula:\nConsider now the exterior product of \\mathbf{v} and \\mathbf{w}:\nwhere the first step uses the distributive law for the [exterior product](https://bliptext.com/articles/), and the last uses the fact that the [exterior product](https://bliptext.com/articles/) is an [alternating map](https://bliptext.com/articles/alternating-map), and in particular (The fact that the [exterior product](https://bliptext.com/articles/) is an [alternating map](https://bliptext.com/articles/alternating-map) also forces ) Note that the coefficient in this last expression is precisely the determinant of the matrix [v w].\nThe fact that this may be positi**v**e or negati**v**e has the intuiti**v**e meaning that **v** and **w** may be oriented in a counterclock**w**ise or clock**w**ise sense as the **v**ertices of the parallelogram they define.\nSuch an area is called the [signed area](https://bliptext.com/articles/signed-area) of the parallelogram: the [absolute value](https://bliptext.com/articles/absolute-value) of the [signed area](https://bliptext.com/articles/signed-area) is the ordinary area, and the sign determines its orientation.\nThe fact that this coefficient is the signed area is not an accident.\nIn 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.\nIn detail, if A(**v**, **w**) denotes the signed area of the parallelogram of **w**hich the pair of **v**ectors **v** and **w** form t**w**o adjacent sides, then A must satisfy the follo**w**ing properties:\nWith the exception of the last property, the exterior product of two vectors satisfies the same properties as the area.\nIn a c****e****rtain s****e****ns****e****, th****e**** ****e****xt****e****rior product g****e****n****e****raliz****e****s th****e**** final prop****e****rty by allowing th****e**** ar****e****a of a parall****e****logram to b****e**** compar****e****d to that of any chos****e****n parall****e****logram in a parall****e****l plan****e**** (h****e****r****e****, th****e**** on****e**** with sid****e****s ****e****1 and ****e****2).\nIn other words, the exterior product provides a basis-independent formulation of area.\n# [The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices. | upload.wikimedia.org/wikipedia/commons/a/ad/Area///parallellogram///as///determinant.svg]\n# [The cross product ( blue vector) in relation to the exterior product ( light blue parallelogram). The length of the cross product is to the length of the parallel unit vector ( red ) as the size of the exterior product is to the size of the reference parallelogram ( light red ). | upload.wikimedia.org/wikipedia/commons/d/d8/Exterior///calc///cross///product.svg]\n\n## Cross and triple products\nFor vectors in **R**3, the exterior algebra is closely related to the [cross product](https://bliptext.com/articles/cross-product) and [triple product](https://bliptext.com/articles/triple-product).\nUsing the standard basis, the exterior product of a pair of vectors\nand\nis\nwhere is the basis for the three-dimensional space ⋀2(**R**3).\nThe coefficients above are the same as those in the usual definition of the [cross product](https://bliptext.com/articles/cross-product) of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a [bivector](https://bliptext.com/articles/bivector).\nBringing in a third vector\nth******e****** ******e******xt******e******rior product of thr******e************e****** v******e******ctors is wh******e******r******e****** ******e******1 ∧ ******e******2 ∧ ******e******3 is th******e****** basis v******e******ctor for th******e****** on******e******-dim******e******nsional spac******e****** ⋀3(**R**3).\nThe scalar coefficient is the [triple product](https://bliptext.com/articles/triple-product) of the three vectors.\nThe cross product and triple product in three dimensions each admit both geometric and algebraic interpretations.\nThe cross prod**u**ct **u** × **v** can be interpreted as a **v**ector which is perpendic**u**lar to both **u** and **v** and whose magnit**u**de is eq**u**al to the area of the parallelogram determined by the two **v**ectors.\nIt can also be interpreted as the **v**ector consisting of the [minors](https://bliptext.com/articles/minor-mathematics) of the matrix with col**u**mns **u** and **v**.\nThe triple prod**u**ct of **u**, **v**, and **w** is geometrically a (signed) **v**ol**u**me.\nAlgebraically, it is the determinant of the matrix **w**ith col**u**mns **u**, **v**, and **w**.\nThe exterior product in three dimensions allows for similar interpretations.\nIn fact, in the presence of a positively oriented [orthonormal basis](https://bliptext.com/articles/orthonormal-basis), the exterior product generalizes these notions to higher dimensions.\n# [Basis Decomposition of a 2-vector | upload.wikimedia.org/wikipedia/commons/8/85/2-vector///decomposition.png]\n\n# Formal definition\nThe exterior algebra of a vector space V over a [field](https://bliptext.com/articles/field-mathematics) K is defined as the [quotient algebra](https://bliptext.com/articles/quotient-ring) of the [tensor algebra](https://bliptext.com/articles/tensor-algebra) T(V), where\nby the two-sided [ideal](https://bliptext.com/articles/ideal-ring-theory) I generated by all elements of the form x \\otimes x such that x\\in V. Symbolically,\nThe exterior product \\wedge of two elements of is defined by\n\n# Algebraic properties\n\n## Alternating product\nThe exterior product is by construction alternating on elements of V, which means that for all x \\in V, by the above construction.\nIt follows that the product is also [anticommutative](https://bliptext.com/articles/anticommutative) on elements of V, for supposing that x, y \\in V, hence\nMore generally, if \\sigma is a [permutation](https://bliptext.com/articles/permutation-group) of the integers [1, \\dots, k], and x_1, x_2, ..., x_k are elements of V, it follows that\nwhere is the [signature of the permutation](https://bliptext.com/articles/signature-of-a-permutation) \\sigma.\nIn particular, if x_i = x_j for some i \\ne j, then the following generalization of the alternating property also holds:\nTogether with the distributive property of the exterior product, one further generalization is that a necessary and sufficient condition for to be a linearly dependent set of vectors is that\n\n## Exterior power\nThe\nk\nth **exterior power** of V, denoted {\\textstyle\\bigwedge}^{\\!k}(V), is the [vector subspace](https://bliptext.com/articles/vector-subspace) of {\\textstyle\\bigwedge}(V) [spanned](https://bliptext.com/articles/linear-span) by elements of the form\nIf \\alpha \\in {\\textstyle\\bigwedge}^{\\!k}(V), then \\alpha is said to be a '''[[p-vector|\nk\n-vector]]'''.\nIf, furthermore, \\alpha can be expressed as an exterior product of k elements of V, then \\alpha is said to be **decomposable** (or **simple**, by some authors; or a **blade**, by others).\nAlthough decomposable k-vectors span {\\textstyle\\bigwedge}^{\\!k}(V), not every element of is decomposable.\nFor example, given \\mathbf{R}^4 with a basis \\{ e_1, e_2, e_3, e_4 \\}, the following 2-vector is not decomposable:\n\n### Basis and dimension\nIf the [dimension](https://bliptext.com/articles/dimension-linear-algebra) of V is n and is a [basis](https://bliptext.com/articles/basis-linear-algebra) for V, then the set\nis a basis for {\\textstyle\\bigwedge}^{\\!k}(V).\nThe reason is the following: given any exterior product of the form\nevery vector v_j can be written as a [linear combination](https://bliptext.com/articles/linear-combination) of the basis vectors e_i; using the bilinearity of the exterior product, this can be expanded to a [linear combination](https://bliptext.com/articles/linear-combination) of exterior products of those basis vectors.\nAny exterior product in which the same basis vector appears more than once is zero; any exterior 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.\nIn general, the resulting coefficients of the basis k-vectors can be computed as the [minors](https://bliptext.com/articles/minor-linear-algebra) of the [matrix](https://bliptext.com/articles/matrix-mathematics) that describes the vectors v_j in terms of the basis e_i.\nBy counting the basis elements, the dimension of is equal to a [binomial coefficient](https://bliptext.com/articles/binomial-coefficient):\nwhere n is the dimension of the vectors, and k is the number of vectors in the product.\nThe binomial coefficient produces the correct result, even for exceptional cases; in particular, for k > n.\nAny element of the exterior algebra can be written as a sum of [[p-vector|\nk\n-vector]]s.\nHence, as a vector space the exterior algebra is a [direct sum](https://bliptext.com/articles/direct-sum-of-modules)\n(where, by convention, {\\textstyle\\bigwedge}^{\\!0}(V) = K, the [field](https://bliptext.com/articles/field-mathematics) underlying V, and {\\textstyle\\bigwedge}^{\\!1}(V) = V), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2^n.\n\n### Rank of a k-vector\nIf \\alpha \\in {\\textstyle\\bigwedge}^{\\!k}(V), then it is possible to express \\alpha as a linear combination of decomposable [[p-vector|\nk\n-vector]]s:\nwhere each is decomposable, say\nThe **rank** of the\nk\n-vector \\alpha is the minimal number of decomposable\nk\n-vectors in such an expansion of \\alpha.\nThis is similar to the notion of [tensor rank](https://bliptext.com/articles/tensor-rank).\nRank is particularly important in the study of 2-vectors.\nThe rank of a 2-vector \\alpha can be identified with half the [rank of the matrix](https://bliptext.com/articles/rank-of-a-matrix) of coefficients of \\alpha in a basis.\nThus if e_i is a basis for V, then \\alpha can be expressed uniquely as where (the matrix of coefficients is [skew-symmetric](https://bliptext.com/articles/skew-symmetric-matrix)).\nThe rank of the matrix a_{ij} is therefore even, and is twice the rank of the form \\alpha.\nIn characteristic 0, the 2-vector \\alpha has rank p if and only if\n\n## Graded structure\nThe exterior product of a\nk\n-vector with a\np\n-vector is a (k + p)-vector, once again invoking bilinearity.\nAs a consequence, the direct sum decomposition of the preceding section\ngives the exterior algebra the additional structure of a [graded algebra](https://bliptext.com/articles/graded-algebra), that is\nMoreover, if\nK\nis the base field, we have\nThe exterior product is graded anticommutative, meaning that if and \\beta \\in {\\textstyle\\bigwedge}^{\\!p}(V), then\nIn addition to studying the graded structure on the exterior algebra, studies additional graded structures on exterior algebras, such as those on the exterior algebra of a [graded module](https://bliptext.com/articles/graded-module) (a module that already carries its own gradation).\n\n## Universal property\nLet\nV\nbe a vector space over the field\nK\n. Informally, multiplication in is performed by manipulating symbols and imposing a [distributive law](https://bliptext.com/articles/distributive-law), an [associative law](https://bliptext.com/articles/associative-law), and using the identity for\nv ∈ V\n. Formally, is the \"most general\" algebra in which these rules hold for the multiplication, in the sense that any unital associative\nK\n-algebra containing\nV\nwith alternating multiplication on\nV\nmust contain a homomorphic image of {\\textstyle\\bigwedge}(V).\nIn other words, the exterior algebra has the following [universal property](https://bliptext.com/articles/universal-property):\nGiven any unital associative\nK\n-algebra\nA\nand any\nK\n-[linear map](https://bliptext.com/articles/linear-map) j : V \\to A such that for every\nv\nin\nV\n, then there exists precisely one unital [algebra homomorphism](https://bliptext.com/articles/algebra-homomorphism) such that\nj(v) = f(i(v))\nfor all\nv\nin\nV\n(here\ni\nis the natural inclusion of\nV\nin {\\textstyle\\bigwedge}(V), see above).\nTo construct the most general algebra that contains\nV\nand whose multiplication is alternating on\nV\n, it is natural to start with the most general associative algebra that contains\nV\n, the [tensor algebra](https://bliptext.com/articles/tensor-algebra)\nT(V)\n, and then enforce the alternating property by taking a suitable [quotient](https://bliptext.com/articles/quotient-ring).\nWe thus take the two-sided [ideal](https://bliptext.com/articles/ideal-ring-theory)\nI\nin\nT(V)\ngenerated by all elements of the form\nv ⊗ v\nfor\nv\nin\nV\n, and define as the quotient\n(and use\n∧\nas the symbol for multiplication in {\\textstyle\\bigwedge}(V)). It is then straightforward to show that contains\nV\nand satisfies the above universal property.\nAs a consequence of this construction, the operation of assigning to a vector space\nV\nits exterior algebra is a [functor](https://bliptext.com/articles/functor) from the [category](https://bliptext.com/articles/category-mathematics) of vector spaces to the [category](https://bliptext.com/articles/category-mathematics) of algebras.\nRather than defining first and then identifying the exterior powers as certain subspaces, one may alternatively define the spaces first and then combine them to form the algebra {\\textstyle\\bigwedge}(V).\nThis approach is often used in differential geometry and is described in the next section.\n# [Universal property of the exterior algebra | upload.wikimedia.org/wikipedia/commons/6/6b/ExteriorAlgebra-01.png]\n\n## Generalizations\nGiven a [commutative ring](https://bliptext.com/articles/commutative-ring) R and an R-[module](https://bliptext.com/articles/module-mathematics) M, we can define the exterior algebra just as above, as a suitable quotient of the tensor algebra \\mathrm{T}(M).\nIt will satisfy the analogous universal property.\nMany of the properties of also require that M be a [projective module](https://bliptext.com/articles/projective-module).\nWhere finite dimensionality is used, the properties further require that M be [finitely generated](https://bliptext.com/articles/finitely-generated-module) and projective.\nGeneralizations to the most common situations can be found in.\nExterior algebras of [vector bundles](https://bliptext.com/articles/vector-bundle) are frequently considered in geometry and topology.\nThere are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the [Serre–Swan theorem](https://bliptext.com/articles/serre-swan-theorem).\nMore general exterior algebras can be defined for [sheaves](https://bliptext.com/articles/sheaf-mathematics) of modules.\n\n# Alternating tensor algebra\nFor a field of characteristic not 2, the exterior algebra of a vector space V over K can be canonically identified with the vector subspace of that consists of [antisymmetric tensors](https://bliptext.com/articles/antisymmetric-tensor).\nFor characteristic 0 (or higher than \\dim V), the vector space of k-linear antisymmetric tensors is transversal to the ideal I, hence, a good choice to represent the quotient.\nBut for nonzero characteristic, the vector space of K-linear antisymmetric tensors could be not transversal to the ideal (actually, for k \\geq \\operatorname{char} K, the vector space of K-linear antisymmetric tensors is contained in I); nevertheless, transversal or not, a product can be defined on this space such that the resulting algebra is isomorphic to the exterior algebra: in the first case the natural choice for the product is just the quotient product (using the available projection), in the second case, this product must be slightly modified as given below (along Arnold setting), but such that the algebra stays isomorphic with the exterior algebra, i.e. the quotient of by the ideal I generated by elements of the form x \\otimes x. Of course, for characteristic 0 (or higher than the dimension of the vector space), one or the other definition of the product could be used, as the two algebras are isomorphic (see V. I. Arnold or Kobayashi-Nomizu).\nLet be the space of homogeneous tensors of degree r. This is spanned by decomposable tensors\nThe **antisymmetrization** (or sometimes the **skew-symmetrization**) of a decomposable tensor is defined by and, when r! \\neq 0 (for nonzero characteristic field r! might be 0):\nwhere the sum is taken over the [symmetric group](https://bliptext.com/articles/symmetric-group) of permutations on the symbols \\{ 1, \\dots, r \\}.\nThis extends by linearity and homogeneity to an operation, also denoted by \\mathcal{A} and \\rm{Alt}, on the full tensor algebra \\mathrm{T}(V).\nNote that\nSuch that, when defined, is the projection for the exterior (quotient) algebra onto the r-homogeneous alternating tensor subspace.\nOn the other hand, the image is always the **alternating tensor graded subspace** (not yet an algebra, as product is not yet defined), denoted A(V).\nThis is a vector subspace of \\mathrm{T}(V), and it inherits the structure of a graded vector space from that on \\mathrm{T}(V).\nMoreover, the kernel of is precisely I^{(r)}, the homogeneous subset of the ideal I, or the kernel of \\mathcal{A} is I. When is defined, A(V) carries an associative graded product defined by (the same as the wedge product)\nAssuming K has characteristic 0, as A(V) is a supplement of I in \\mathrm{T}(V), with the above given product, there is a canonical isomorphism\nWhen the characteristic of the field is nonzero, \\mathcal{A} will do what \\rm{Alt} did before, but the product cannot be defined as above.\nIn such a case, isomorphism still holds, in spite of A(V) not being a supplement of the ideal I, but then, the product should be modified as given below ( product, Arnold setting).\nFinally, we always get A(V) isomorphic with {\\textstyle\\bigwedge}(V), but the product could (or should) be chosen in two ways (or only one).\nActually, the product could be chosen in many ways, rescaling it on homogeneous spaces as for an arbitrary sequence c(r) in the field, as long as the division makes sense (this is such that the redefined product is also associative, i.e. defines an algebra on A(V)).\nAlso note, the interior product definition should be changed accordingly, in order to keep its skew derivation property.\n\n## Index notation\nSuppose that V has finite dimension n, and that a basis e1, ..., en of V is given.\nThen any alternating tensor t ∈ Ar(V) ⊂ Tr(V) can be written in [index notation](https://bliptext.com/articles/index-notation) with the [Einstein summation convention](https://bliptext.com/articles/einstein-summation-convention) as where ti1⋅⋅⋅ir is [completely antisymmetric](https://bliptext.com/articles/antisymmetric-tensor) in its indices.\nThe exterior product of two alternating tensors t and s of ranks r and p is given by\nThe components of this tensor are precisely the skew part of the components of the tensor product s ⊗ t, denoted by square brackets on the indices:\nThe [interior product](https://bliptext.com/articles/) may also be described in index notation as follows.\nLet be an antisymmetric tensor of rank r. Then, for α ∈ V∗, \\iota_\\alpha t is an alternating tensor of rank r - 1, given by where n is the dimension of V.\n\n# Duality\n\n## Alternating operators\nGiven two vector spaces V and X and a natural number k, an **alternating operator** from Vk to X is a [multilinear map](https://bliptext.com/articles/multilinear-map)\nsuch that whenever v1, ..., vk are [linearly dependent](https://bliptext.com/articles/linearly-dependent) vectors in V, then\nThe map\nwhich associates to k vectors from V their exterior product, i.e. their corresponding k-vector, is also alternating.\nIn fact, this map is the \"most general\" alternating operator defined on V^k; given any other alternating operator there exists a unique [linear map](https://bliptext.com/articles/linear-map) with This [universal property](https://bliptext.com/articles/universal-property) characterizes the space of alternating operators on V^k and can serve as its definition.\n\n## Alternating multilinear forms\n[[File:N-form.svg|thumb|upright=0.6|Geometric interpretation for the **exterior product** of\nn\n[1-forms](https://bliptext.com/articles/1-form) (\nε\n,\nη\n,\nω\n) to obtain an\nn\n-form (\"mesh\" of [coordinate surfaces](https://bliptext.com/articles/coordinate-surface), here planes), for\nn = 1, 2, 3\n. The \"circulations\" show [orientation](https://bliptext.com/articles/orientation-vector-space).\n]]\nThe above discussion specializes to the case when X = K, the base field.\nIn this case an alternating multilinear function\nis called an **alternating multilinear form**.\nThe set of all [alternating](https://bliptext.com/articles/alternating-map) [multilinear forms](https://bliptext.com/articles/multilinear-form) is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again [alternating](https://bliptext.com/articles/alternating-map).\nBy the universal property of the exterior power, the space of alternating forms of degree k on V is [naturally](https://bliptext.com/articles/natural-transformation) isomorphic with the [dual vector space](https://bliptext.com/articles/dual-vector-space) \\bigl({\\textstyle\\bigwedge}^{\\!k}(V)\\bigr)^*.\nIf V is finite-dimensional, then the latter is to {\\textstyle\\bigwedge}^{\\!k}\\left(V^*\\right).\nIn particular, if V is n-dimensional, the dimension of the space of alternating maps from V^k to K is the [binomial coefficient](https://bliptext.com/articles/binomial-coefficient) \\textstyle\\binom{n}{k}.\nUnder such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones.\nSuppose ω : Vk → K and η : Vm → K are two anti-symmetric maps.\nAs in the case of [tensor products](https://bliptext.com/articles/tensor-product) of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables.\nDepending on the choice of identification of elements of exterior power with multilinear forms, the exterior product is defined as or as where, if the characteristic of the base field K is 0, the alternation Alt of a multilinear map is defined to be the average of the sign-adjusted values over all the [permutations](https://bliptext.com/articles/permutation) of its variables:\nWhen the [field](https://bliptext.com/articles/field-mathematics) K has [finite characteristic](https://bliptext.com/articles/characteristic-of-a-field), an equivalent version of the second expression without any factorials or any constants is well-defined:\nwhere here Shk,m ⊂ Sk+m is the subset of [[(p,q) shuffle|\n(k, m)\nshuffles]]: [permutations](https://bliptext.com/articles/permutation) σ of the set such that σ(1) < σ(2) < ⋯ < σ(k), and σ(k + 1) < σ(k + 2) < ... < σ(k + m).\nAs this might look very specific and fine tuned, an equivalent raw version is to sum in the above formula over permutations in left cosets of Sk+m / (Sk × Sm).\n\n## Interior product\nSuppose that V is finite-dimensional.\nIf V^* denotes the [dual space](https://bliptext.com/articles/dual-space) to the vector space V, then for each \\alpha \\in V^*, it is possible to define an [antiderivation](https://bliptext.com/articles/derivation-abstract-algebra) on the algebra {\\textstyle\\bigwedge}(V),\nThis derivation is called the **interior product** with \\alpha, or sometimes the **insertion operator**, or **contraction** by \\alpha.\nSuppose that w \\in {\\textstyle\\bigwedge}^{\\!k}(V).\nThen w is a multilinear mapping of V^* to K, so it is defined by its values on the\nk\n-fold [Cartesian product](https://bliptext.com/articles/cartesian-product) V^* \\times V^* \\times \\dots \\times V^*.\nIf u1, u2, ..., uk−1 are k - 1 elements of V^*, then define\nAdditionally, let whenever f is a pure scalar (i.e., belonging to {\\textstyle\\bigwedge}^{\\!0}(V)).\n\n### Axiomatic characterization and properties\nThe interior product satisfies the following properties:\nThese three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.\nFurther properties of the interior product include:\n\n## Hodge duality\nSuppose that V has finite dimension n. Then the interior product induces a canonical isomorphism of vector spaces\nby the recursive definition\nIn the geometrical setting, a non-zero element of the top exterior power (which is a one-dimensional vector space) is sometimes called a **[volume form](https://bliptext.com/articles/volume-form)** (or **orientation form**, although this term may sometimes lead to ambiguity).\nThe name orientation form comes from the fact that a choice of preferred top element determines an orientation of the whole exterior algebra, since it is tantamount to fixing an ordered basis of the vector space.\nRelative to the preferred volume form \\sigma, the isomorphism is given explicitly by\nIf, in addition to a volume form, the vector space V is equipped with an [inner product](https://bliptext.com/articles/inner-product) identifying V with V^*, then the resulting isomorphism is called the **Hodge star operator**, which maps an element to its **Hodge dual**:\nThe composition of \\star with itself maps and is always a scalar multiple of the identity map.\nIn most applications, the volume form is compatible with the inner product in the sense that it is an exterior product of an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) of V. In this case,\nwhere id is the identity mapping, and the inner product has [metric signature](https://bliptext.com/articles/metric-signature) (p, q) — p pluses and q minuses.\n\n## Inner product\nFor V a finite-dimensional space, an [inner product](https://bliptext.com/articles/inner-product) (or a [pseudo-Euclidean](https://bliptext.com/articles/pseudo-euclidean-space) [inner product](https://bliptext.com/articles/inner-product)) on V defines an isomorphism of V with V^*, and so also an isomorphism of with \\bigl({\\textstyle\\bigwedge}^{\\!k} V\\bigr)^*.\nThe pairing between these two spaces also takes the form of an inner product.\nOn decomposable k-vectors,\nthe determinant of the matrix of inner products.\nIn the special case vi = wi, the inner product is the square norm of the k-vector, given by the determinant of the [Gramian matrix](https://bliptext.com/articles/gramian-matrix) (⟨vi, vj⟩).\nThis is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on If ei, i = 1, 2, ..., n, form an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) of V, then the vectors of the form\nconstitute an orthonormal basis for {\\textstyle\\bigwedge}^{\\!k}(V), a statement equivalent to the [Cauchy–Binet formula](https://bliptext.com/articles/cauchy-binet-formula).\nWith respect to the inner product, exterior multiplication and the interior product are mutually adjoint.\nSpecifically, for \\mathbf{v}\\in {\\textstyle\\bigwedge}^{\\!k-1}(V), \\mathbf{w} \\in {\\textstyle\\bigwedge}^{\\!k}(V), and x \\in V, where x♭ ∈ V∗ is the [musical isomorphism](https://bliptext.com/articles/musical-isomorphism), the linear functional defined by for all y \\in V. This property completely characterizes the inner product on the exterior algebra.\nIndeed, more generally for \\mathbf{v}\\in {\\textstyle\\bigwedge}^{\\!k-l}(V), \\mathbf{w}\\in {\\textstyle\\bigwedge}^{\\!k}(V), and \\mathbf{x}\\in {\\textstyle\\bigwedge}^{\\!l}(V), iteration of the above adjoint properties gives\nwhere now is the dual l-vector defined by for all \\mathbf{y} \\in {\\textstyle\\bigwedge}^{\\!l}(V).\n\n## Bialgebra structure\nThere is a correspondence between the graded dual of the graded algebra and alternating multilinear forms on V. The exterior algebra (as well as the [symmetric algebra](https://bliptext.com/articles/symmetric-algebra)) inherits a bialgebra structure, and, indeed, a [Hopf algebra](https://bliptext.com/articles/hopf-algebra) structure, from the [tensor algebra](https://bliptext.com/articles/tensor-algebra).\nSee the article on [tensor algebras](https://bliptext.com/articles/tensor-algebra) for a detailed treatment of the topic.\nThe exterior product of multilinear forms defined above is dual to a [coproduct](https://bliptext.com/articles/coproduct) defined on {\\textstyle\\bigwedge}(V), giving the structure of a [coalgebra](https://bliptext.com/articles/coalgebra).\nThe **coproduct** is a linear function \\Delta : {\\textstyle\\bigwedge}(V) \\to {\\textstyle\\bigwedge}(V) \\otimes {\\textstyle\\bigwedge}(V), which is given by on elements v \\in V. The symbol 1 stands for the unit element of the field K. Recall that K \\simeq {\\textstyle\\bigwedge}^{\\!0}(V) \\subseteq {\\textstyle\\bigwedge}(V), so that the above really does lie in {\\textstyle\\bigwedge}(V) \\otimes {\\textstyle\\bigwedge}(V).\nThis definition of the coproduct is lifted to the full space by (linear) homomorphism.\nThe correct form of this homomorphism is not what one might naively write, but has to be the one carefully defined in the [coalgebra](https://bliptext.com/articles/coalgebra) article.\nIn this case, one obtains\nExpanding this out in detail, one obtains the following expression on decomposable elements:\nwhere the second summation is taken over all [[(p,q) shuffle|\n(p, k−p)\n-shuffles]].\nBy convention, one takes that Sh(k,0) and Sh(0,k) equals {id: {1, ..., k} → {1, ..., k}}.\nIt is also convenient to take the pure wedge products and\nto equal 1 for p = 0 and p = k, respectively (the empty product in ).\nThe shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements x_k is preserved in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.\nObserve that the coproduct preserves the grading of the algebra.\nExtending to the full space one has\nThe tensor symbol ⊗ used in this section should be understood with some caution: it is not the same tensor symbol as the one being used in the definition of the alternating product.\nIntuitively, it is perhaps easiest to think it as just another, but different, tensor product: it is still (bi-)linear, as tensor products should be, but it is the product that is appropriate for the definition of a bialgebra, that is, for creating the object {\\textstyle\\bigwedge}(V) \\otimes {\\textstyle\\bigwedge}(V).\nAny lingering doubt can be shaken by pondering the equalities (1 ⊗ v) ∧ (1 ⊗ w) = 1 ⊗ (v ∧ w) and (v ⊗ 1) ∧ (1 ⊗ w) = v ⊗ w, which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols.\nThis distinction is developed in greater detail in the article on [tensor algebras](https://bliptext.com/articles/tensor-algebra).\nHere, there is much less of a problem, in that the alternating product \\wedge clearly corresponds to multiplication in the exterior algebra, leaving the symbol \\otimes free for use in the definition of the bialgebra.\nIn practice, this presents no particular problem, as long as one avoids the fatal trap of replacing alternating sums of \\otimes by the wedge symbol, with one exception.\nOne can construct an alternating product from \\otimes, with the understanding that it works in a different space.\nImmediately below, an example is given: the alternating product for the dual space can be given in terms of the coproduct.\nThe construction of the bialgebra here parallels the construction in the [tensor algebra](https://bliptext.com/articles/tensor-algebra) article almost exactly, except for the need to correctly track the alternating signs for the exterior algebra.\nIn terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:\nwhere the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely, α ∧ β = ε ∘ (α ⊗ β) ∘ Δ, where \\varepsilon is the counit, as defined presently).\nThe **counit** is the homomorphism that returns the 0-graded component of its argument.\nThe coproduct and counit, along with the exterior product, define the structure of a [bialgebra](https://bliptext.com/articles/bialgebra) on the exterior algebra.\nWith an **antipode** defined on homogeneous elements by S(x) = (-1)^{\\binom{\\text{deg}\\, x\\, + 1}{2} }x, the exterior algebra is furthermore a [Hopf algebra](https://bliptext.com/articles/hopf-algebra).\n\n# Functoriality\nSuppose that V and W are a pair of vector spaces and f : V \\to W is a [linear map](https://bliptext.com/articles/linear-map).\nThen, by the universal property, there exists a unique homomorphism of graded algebras\nsuch that\nIn particular, preserves homogeneous degree.\nThe\nk\n-graded components of are given on decomposable elements by\nLet\nThe components of the transformation {\\textstyle\\bigwedge}^{\\!k}(f) relative to a basis of V and W is the matrix of k \\times k minors of f. In particular, if V = W and V is of finite dimension n, then is a mapping of a one-dimensional vector space to itself, and is therefore given by a scalar: the [determinant](https://bliptext.com/articles/determinant) of f.\n\n## Exactness\nIf is a [short exact sequence](https://bliptext.com/articles/short-exact-sequence) of vector spaces, then\nis an exact sequence of graded vector spaces, as is\n\n## Direct sums\nIn particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:\nThis is a graded isomorphism; i.e.,\nIn greater generality, for a short exact sequence of vector spaces there is a natural [filtration](https://bliptext.com/articles/filtration-mathematics)\nwhere F^p for p \\geq 1 is spanned by elements of the form for u_i \\in U and v_i \\in V. The corresponding quotients admit a natural isomorphism\nIn particular, if U is 1-dimensional then\nis exact, and if W is 1-dimensional then\nis exact.\n\n# Applications\n\n## Oriented volume in affine space\nThe natural setting for (oriented) k-dimensional volume and exterior algebra is [affine space](https://bliptext.com/articles/affine-space).\nThis is also the intimate connection between exterior algebra and [differential forms](https://bliptext.com/articles/differential-forms), as to integrate we need a 'differential' object to measure infinitesimal volume.\nIf \\mathbb{A} is an affine space over the vector space V, and a ([simplex](https://bliptext.com/articles/simplex)) collection of ordered k+1 points, we can define its oriented k-dimensional volume as the exterior product of vectors (using concatenation PQ to mean the [displacement vector](https://bliptext.com/articles/displacement-vector) from point P to Q); if the order of the points is changed, the oriented volume changes by a sign, according to the parity of the permutation.\nIn n-dimensional space, the volume of any n-dimensional simplex is a scalar multiple of any other.\nThe sum of the (k-1)-dimensional oriented areas of the boundary simplexes of a k-dimensional simplex is zero, as for the sum of vectors around a triangle or the oriented triangles bounding the tetrahedron in the previous section.\nThe vector space structure on generalises addition of vectors in V: we have and similarly a\nk\n-blade is linear in each factor.\n\n## Linear algebra\nIn applications to [linear algebra](https://bliptext.com/articles/linear-algebra), the exterior product provides an abstract algebraic manner for describing the [determinant](https://bliptext.com/articles/determinant) and the [minors](https://bliptext.com/articles/minor-[matrix](https://bliptext.com/articles/matrix-mathematics)) of a [matrix](https://bliptext.com/articles/matrix-mathematics).\nFor instance, it is well known that the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix (with a sign to track orientation).\nThis suggests that the determinant can be defined in terms of the exterior product of the column vectors.\nLikewise, the\nk × k\nminors of a matrix can be defined by looking at the exterior products of column vectors chosen\nk\nat a time.\nThese ideas can be extended not just to matrices but to [linear transformations](https://bliptext.com/articles/linear-transformation) as well: the determinant of a linear transformation is the factor by which it scales the oriented volume of any given reference parallelotope.\nSo the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power.\nThe action of a transformation on the lesser exterior powers gives a [basis](https://bliptext.com/articles/basis-of-a-vector-space)-independent way to talk about the minors of the transformation.\n\n## Physics\nIn physics, many quantities are naturally represented by alternating operators.\nFor example, if the motion of a charged particle is described by velocity and acceleration vectors in four-dimensional spacetime, then normalization of the velocity vector requires that the electromagnetic force must be an alternating operator on the velocity.\nIts six degrees of freedom are identified with the electric and magnetic fields.\n\n## Electromagnetic field\nIn [Einstein's theories of relativity](https://bliptext.com/articles/theory-of-relativity), the [electromagnetic field](https://bliptext.com/articles/electromagnetic-field) is generally given as a [differential 2-form](https://bliptext.com/articles/differential-form) F = dA in [4-space](https://bliptext.com/articles/4-space) or as the equivalent [alternating tensor field](https://bliptext.com/articles/antisymmetric-tensor) the [electromagnetic tensor](https://bliptext.com/articles/electromagnetic-tensor).\nThen or the equivalent Bianchi identity\nNone of this requires a metric.\nAdding the [Lorentz metric](https://bliptext.com/articles/lorentz-metric) and an [orientation](https://bliptext.com/articles/orientability) provides the [Hodge star operator](https://bliptext.com/articles/hodge-star-operator) \\star and thus makes it possible to define or the equivalent tensor [divergence](https://bliptext.com/articles/divergence) where\n\n## Linear geometry\nThe decomposable\nk\n-vectors have geometric interpretations: the bivector u \\wedge v represents the plane spanned by the vectors, \"weighted\" with a number, given by the area of the oriented [parallelogram](https://bliptext.com/articles/parallelogram) with sides u and v. Analogously, the 3-vector represents the spanned 3-space weighted by the volume of the oriented [parallelepiped](https://bliptext.com/articles/parallelepiped) with edges u, v, and w.\n\n## Projective geometry\nDecomposable\nk\n-vectors in correspond to weighted\nk\n-dimensional [linear subspaces](https://bliptext.com/articles/linear-subspace) of V. In particular, the [Grassmannian](https://bliptext.com/articles/grassmannian) of\nk\n-dimensional subspaces of V, denoted \\operatorname{Gr}_k(V), can be naturally identified with an [algebraic subvariety](https://bliptext.com/articles/algebraic-variety) of the [projective space](https://bliptext.com/articles/projective-space).\nThis is called the [Plücker embedding](https://bliptext.com/articles/pl-cker-embedding), and the image of the embedding can be characterized by the [Plücker relations](https://bliptext.com/articles/pl-cker-relations).\n\n## Differential geometry\nThe exterior algebra has notable applications in [differential geometry](https://bliptext.com/articles/differential-geometry), where it is used to define [differential forms](https://bliptext.com/articles/differential-form).\nDifferential forms are mathematical objects that evaluate the length of vectors, areas of parallelograms, and volumes of [higher-dimensional bodies](https://bliptext.com/articles/parallelepiped), so they can be [integrated](https://bliptext.com/articles/integral) over curves, surfaces and higher dimensional [manifolds](https://bliptext.com/articles/manifold) in a way that generalizes the [line integrals](https://bliptext.com/articles/line-integral) and [surface integrals](https://bliptext.com/articles/surface-integral) from calculus.\nA [differential form](https://bliptext.com/articles/differential-form) at a point of a [differentiable manifold](https://bliptext.com/articles/differentiable-manifold) is an alternating multilinear form on the [tangent space](https://bliptext.com/articles/tangent-space) at the point.\nEquivalently, a differential form of degree\nk\nis a [linear functional](https://bliptext.com/articles/linear-functional) on the\nk\nth exterior power of the tangent space.\nAs a consequence, the exterior product of multilinear forms defines a natural exterior product for differential forms.\nDifferential forms play a major role in diverse areas of differential geometry.\nAn [alternate approach](https://bliptext.com/articles/differential-mathematics) defines differential forms in terms of [germs of functions](https://bliptext.com/articles/germ-mathematics).\nIn particular, the [exterior derivative](https://bliptext.com/articles/exterior-derivative) gives the exterior algebra of differential forms on a manifold the structure of a [differential graded algebra](https://bliptext.com/articles/differential-graded-algebra).\nThe exterior derivative commutes with [pullback](https://bliptext.com/articles/pullback-differential-geometry) along smooth mappings between manifolds, and it is therefore a [natural](https://bliptext.com/articles/natural-transformation) [differential operator](https://bliptext.com/articles/differential-operator).\nThe exterior algebra of differential forms, equipped with the exterior derivative, is a [cochain complex](https://bliptext.com/articles/cochain-complex) whose cohomology is called the [de Rham cohomology](https://bliptext.com/articles/de-rham-cohomology) of the underlying manifold and plays a vital role in the [algebraic topology](https://bliptext.com/articles/algebraic-topology) of differentiable manifolds.\n\n## Representation theory\nIn [representation theory](https://bliptext.com/articles/representation-theory), the exterior algebra is one of the two fundamental [Schur functors](https://bliptext.com/articles/schur-functor) on the category of vector spaces, the other being the [symmetric algebra](https://bliptext.com/articles/symmetric-algebra).\nTogether, these constructions are used to generate the [irreducible representations](https://bliptext.com/articles/irreducible-representation) of the [general linear group](https://bliptext.com/articles/general-linear-group) (see [Fundamental representation](https://bliptext.com/articles/fundamental-representation)).\n\n## Superspace\nThe exterior algebra over the complex numbers is the archetypal example of a [superalgebra](https://bliptext.com/articles/superalgebra), which plays a fundamental role in physical theories pertaining to [fermions](https://bliptext.com/articles/fermion) and [supersymmetry](https://bliptext.com/articles/supersymmetry).\nA single element of the exterior algebra is called a **supernumber** or [Grassmann number](https://bliptext.com/articles/grassmann-number).\nThe exterior algebra itself is then just a one-dimensional [superspace](https://bliptext.com/articles/superspace): it is just the set of all of the points in the exterior algebra.\nThe topology on this space is essentially the [weak topology](https://bliptext.com/articles/weak-topology), the [open sets](https://bliptext.com/articles/open-sets) being the [cylinder sets](https://bliptext.com/articles/cylinder-set).\nAn\nn\n-dimensional superspace is just the n-fold product of exterior algebras.\n\n## Lie algebra homology\nLet L be a Lie algebra over a field K, then it is possible to define the structure of a [chain complex](https://bliptext.com/articles/chain-complex) on the exterior algebra of L. This is a K-linear mapping\ndefined on decomposable elements by\nThe [Jacobi identity](https://bliptext.com/articles/jacobi-identity) holds if and only if \\partial\\partial = 0, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra L to be a Lie algebra.\nMoreover, in that case is a [chain complex](https://bliptext.com/articles/chain-complex) with boundary operator \\partial.\nThe [homology](https://bliptext.com/articles/homology-theory) associated to this complex is the Lie algebra [homology](https://bliptext.com/articles/homology-theory).\n\n## Homological algebra\nThe exterior algebra is the main ingredient in the construction of the [Koszul complex](https://bliptext.com/articles/koszul-complex), a fundamental object in [homological algebra](https://bliptext.com/articles/homological-algebra).\n\n# History\nThe exterior algebra was first introduced by [Hermann Grassmann](https://bliptext.com/articles/hermann-grassmann) in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension.\nThis referred more generally to an algebraic (or axiomatic) theory of extended quantities and was one of the early precursors to the modern notion of a [vector space](https://bliptext.com/articles/vector-space).\n[Saint-Venant](https://bliptext.com/articles/adh-mar-jean-claude-barr-de-saint-venant) also published similar ideas of exterior calculus for which he claimed priority over Grassmann.\nThe algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors.\nIt was thus a calculus, much like the [propositional calculus](https://bliptext.com/articles/propositional-calculus), except focused exclusively on the task of formal reasoning in geometrical terms.\nIn particular, this new development allowed for an axiomatic characterization of dimension, a property that had previously only been examined from the coordinate point of view.\nThe import of this new theory of vectors and [multivectors](https://bliptext.com/articles/multivector) was lost to mid-19th-century mathematicians,\nuntil being thoroughly vetted by [Giuseppe Peano](https://bliptext.com/articles/giuseppe-peano) in 1888.\nPeano's work also remained somewhat obscure until the turn of the century, when the subject was unified by members of the French geometry school (notably [Henri Poincaré](https://bliptext.com/articles/henri-poincar), [Élie Cartan](https://bliptext.com/articles/lie-cartan), and [Gaston Darboux](https://bliptext.com/articles/gaston-darboux)) who applied Grassmann's ideas to the calculus of [differential forms](https://bliptext.com/articles/differential-form).\nA short while later, [Alfred North Whitehead](https://bliptext.com/articles/alfred-north-whitehead), borrowing from the ideas of Peano and Grassmann, introduced his [universal algebra](https://bliptext.com/articles/universal-algebra).\nThis then paved the way for the 20th-century developments of [abstract algebra](https://bliptext.com/articles/abstract-algebra) by placing the axiomatic notion of an algebraic system on a firm logical footing.\n\n# Mathematical references\n\n## Historical references\n\n## Other references and further reading","revisionCount":0},"hasMore":false,"nextCursor":null}