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Encyclopedia > Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of vectors is used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. In linear algebra, the exterior product provides an abstract algebraic basis-independent manner for describing the determinant and the minors of a linear transformation, and is fundamentally related to ideas of rank and linear independence. The exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann[1]) of a given vector space V is the algebra generated by the exterior product. It is widely used in contemporary geometry, especially differential geometry and algebraic geometry through the algebra of differential forms, as well as in multilinear algebra and related fields. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... For the crossed product in algebra and functional analysis, see crossed product. ... In physics and engineering, a vector is a physical entity which has a magnitude which is a scalar (a physical quantity expressed as the product of a numerical value and a physical unit, not just a number). ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ... Hermann GÃ¼nther Grassmann (April 15, 1809, Stettin â€“ September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, multilinear algebra extends the methods of linear algebra. ...

Formally, the exterior algebra is a certain unital associative algebra over a field K which contains V as a subspace. It is denoted by Λ(V) or Λ(V) and its multiplication is also known as the wedge product or the exterior product and is written as $wedge$. The wedge product is an associative and bilinear operation ∧ : Λ(V) × Λ(V) → Λ(V). Its essential feature is that it is alternating on V: In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

(1) $vwedge v = 0 mbox{ for all }vin V,$

which implies in particular

(2) $uwedge v = - vwedge u$ for all $u,vin V$, and
(3) $v_1wedge v_2wedgecdots wedge v_k = 0$ whenever $v_1, ldots, v_k in V$ are linearly dependent.[2]

In terms of category theory, the exterior algebra is type of functor on vector spaces, given by a universal construction. It is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the wedge product. This dual algebra is precisely the algebra of alternating multilinear forms on V, and the pairing between the exterior algebra and its dual is given by the interior product. With the additional structure of a volume form, the exterior algebra becomes a Hopf algebra whose antipode is the Hodge dual. In many cases, the exterior algebra is naturally realized as a certain subspace of the tensor algebra of V. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In mathematics, a bialgebra over a field K is a structure which is both a unital associative algebra and a coalgebra over K, such that the comultiplication and the counit are both unital algebra homomorphisms. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ... In multilinear algebra, a multilinear form is a map of the type , where V is a vector space over the field K, that is separately linear in each its N variables. ... In mathematics, the interior product is a degree âˆ’1 derivation on the exterior algebra of differential forms on a smooth manifold. ... In mathematics, a volume form is a nowhere zero differential n-form on an oriented n-manifold. ... In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Î” is the comultiplication of the bialgebra, âˆ‡ its multiplication, Î· its unit and Îµ its counit. ... In mathematics, the Hodge star operator or Hodge dual is a signficant linear map introduced in general by W. V. D. Hodge. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ...

## Motivating examples GA_googleFillSlot("encyclopedia_square");

### Areas in the plane

The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices.

The Cartesian plane R2 is a vector space equipped with a basis consisting of a pair of unit vectors Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

${mathbf e}_1 = (1,0),quad {mathbf e}_2 = (0,1).$

Suppose that

$v = v_1{mathbf e}_1 + v_2{mathbf e}_2, quad w = w_1{mathbf e}_1 + w_2{mathbf e}_2$

are a pair of given vectors in R2, written in components. There is a unique parallelogram having v and w as two of its sides. The area of this parallelogram is given by the standard determinant formula: In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

$A = left|detbegin{bmatrix}v& wend{bmatrix}right| = |v_1w_2 - v_2w_1|.$

Consider now the exterior product of v and w:

$vwedge w = (v_1{mathbf e}_1 + v_2{mathbf e}_2)wedge (w_1{mathbf e}_1 + w_2{mathbf e}_2)=v_1w_1{mathbf e}_1wedge{mathbf e}_1+ v_1w_2{mathbf e}_1wedge {mathbf e}_2+v_2w_1{mathbf e}_2wedge {mathbf e}_1+v_2w_2{mathbf e}_2wedge {mathbf e}_2$
$=(v_1w_2-v_2w_1){mathbf e}_1wedge{mathbf e}_2$

where the first step uses the distributive law for the wedge product, and the last uses the fact that the wedge product is alternating. Note that the coefficient in this last expression is precisely the determinant of the matrix [v w]. The fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the signed area of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation.

The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if A(v,w) denotes the signed area of the parallelogram determined by the pair of vectors v and w, then A must satisfy the following properties:

1. A(av,bw) = a b A(v,w) for any real numbers a and b, since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram).
2. A(v,v) = 0, since the area of the degenerate parallelogram determined by v (i.e., a line segment) is zero.
3. A(w,v) = -A(v,w), since interchanging the roles of v and w reverses the orientation of the parallelogram.
4. A(v + aw,w) = A(v,w), since adding a multiple of w to v affects neither the base nor the height of the parallelogram and consequently preserves its area.
5. A(e1, e2) = 1, since the area of the unit square is one.

With the exception of the last property, the wedge product satisfies the same formal properties as the area. In a certain sense, the wedge product generalizes the final property by allowing the area of a parallelogram to be compared to that of any "standard" chosen parallelogram. In other words, the exterior product in two-dimensions is a basis-independent formulation of area.[3] In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ...

### Cross and triple products

For vectors in R3, the exterior algebra is closely related to the cross product and triple product. Using the standard basis {e1, e2, e3}, the wedge product of a pair of vectors For the crossed product in algebra and functional analysis, see crossed product. ... This article is about mathematics. ...

$mathbf{u} = u_1 mathbf{e}_1 + u_2 mathbf{e}_2 + u_3 mathbf{e}_3$

and

$mathbf{v} = v_1 mathbf{e}_1 + v_2 mathbf{e}_2 + v_3 mathbf{e}_3$

is

$mathbf{u} wedge mathbf{v} = (u_1 v_2 - u_2 v_1) (mathbf{e}_1 wedge mathbf{e}_2) + (u_1 v_3 - u_3 v_1) (mathbf{e}_1 wedge mathbf{e}_3) + (u_2 v_3 - u_3 v_2) (mathbf{e}_2 wedge mathbf{e}_3)$

where {e1 Λ e2, e1 Λ e3, e2 Λ e3} is the basis for the three-dimensional space Λ2(R3). This imitates the usual definition of the cross product of vectors in three dimensions. For the crossed product in algebra and functional analysis, see crossed product. ...

Bringing in a third vector

$mathbf{w} = w_1 mathbf{e}_1 + w_2 mathbf{e}_2 + w_3 mathbf{e}_3$,

the wedge product of three vectors is

$mathbf{u} wedge mathbf{v} wedge mathbf{w} = (u_1 v_2 w_3 + u_2 v_3 w_1 + u_3 v_1 w_2 - u_1 v_3 w_2 - u_2 v_1 w_3 - u_3 v_2 w_1) (mathbf{e}_1 wedge mathbf{e}_2 wedge mathbf{e}_3)$

where e1 Λ e2 Λ e3 is the basis vector for the one-dimensional space Λ3(R3). This imitates the usual definition of the triple product. This article is about mathematics. ...

The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product u×v can be interpreted as a vector which is perpendicular to both u and v and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the minors of the matrix with columns u and v. The triple product of u, v, and w is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns u, v, and w. The exterior product in three-dimensions allows for similar interpretations. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions. In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ... In mathematics, an orthonormal basis of an inner product space V(i. ...

## Formal definitions and algebraic properties

The exterior algebra Λ(V) over a vector space V is defined as the quotient algebra of the tensor algebra by the two-sided ideal I generated by all elements of the form $x otimes x$ such that xV. Symbolically, In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In mathematics, the term ideal has multiple meanings. ...

$Lambda(V) := T(V)/(x otimes x).$

The wedge product ∧ of two elements of Λ(V) is defined by

$alphawedgebeta = alphaotimesbeta pmod I.$

### Anticommutativity of the wedge product

This product is anticommutative on elements of V, for supposing that x, yV, A mathematical operator (typically a binary operator, represented by *) is anticommutative iff it is true that x * y = &#8722;(y * x) for all x and y on the operators valid domain (e. ...

$0 equiv (x+y)wedge (x+y) = xwedge x + xwedge y + ywedge x + ywedge y equiv xwedge y + ywedge x pmod I$

whence

$xwedge y = - ywedge x.$

More generally, if x1x2, ..., xk are elements of V, and σ is a permutation of the integers [1,...,k], then In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ...

$x_{sigma(1)}wedge x_{sigma(2)}wedgedotswedge x_{sigma(k)} = {rm sgn}(sigma)x_1wedge x_2wedgedots x_k,$

where sgn(σ) is the signature of the permutation σ. In mathematics, the permutations of a finite set (i. ...

### The exterior power

The k-th exterior power of V, denoted Λk(V), is the vector subspace of Λ(V) spanned by elements of the form The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ...

$x_1wedge x_2wedgedotswedge x_k,quad x_iin V, i=1,2,dots, k.$

If α ∈ Λk(V), then α is said to be a k-multivector. If, furthermore, α can be expressed as a wedge product of k elements of V, then α is said to be decomposable. Although decomposable multivectors span Λk(V), not every element of Λk(V) is decomposable. For example, in R4, the following 2-multivector is not decomposable:

$alpha = e_1wedge e_2 + e_3wedge e_4.$

(This is in fact a symplectic form. To see this, one need only check that α ∧ α ≠ 0.) In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

#### Basis and dimension

If the dimension of V is n and {e1,...,en} is a basis of V, then the set In mathematics, the dimension of a vector space V is the cardinality (i. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

${e_{i_1}wedge e_{i_2}wedgecdotswedge e_{i_k} mid 1le i_1 < i_2 < cdots < i_k le n}$

is a basis for Λk(V). The reason is the following: given any wedge product of the form

$v_1wedgecdotswedge v_k$

then every vector vj can be written as a linear combination of the basis vectors ei; using the bilinearity of the wedge product, this can be expanded to a linear combination of wedge products of those basis vectors. Any wedge product in which the same basis vector appears more than once is zero; any wedge product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis k-vectors can be computed as the minors of the matrix that describes the vectors vj in terms of the basis ei. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

Counting the basis elements, we see that the dimension of Λk(V) is the binomial coefficient n choose k. In particular, Λk(V) = {0} for k > n. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...

Any element of the exterior algebra can be written as a sum of multivectors. Hence, as a vector space the exterior algebra is a direct sum In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

$Lambda(V) = Lambda^0(V)oplus Lambda^1(V) oplus Lambda^2(V) oplus cdots oplus Lambda^n(V)$

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

### Graded structure

The wedge product of a k-multivector with a p-multivector is a (k+p)-multivector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section

$Lambda(V) = Lambda^0(V)oplus Lambda^1(V) oplus Lambda^2(V) oplus cdots oplus Lambda^n(V)$

gives the exterior algebra the additional structure of a graded algebra. Symbolically, In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ...

$left(Lambda^k(V)right)wedgeleft(Lambda^p(V)right)sub Lambda^{k+p}(V).$

Moreover, the wedge product is graded anticommutative, meaning that if α ∈ Λk(V) and β ∈ Λp(V), then

$alphawedgebeta = (-1)^{kp}betawedgealpha.$

### Universal property

Let V be a vector space over the field K. Informally, multiplication in Λ(V) is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identities vv = 0 for vV and vw = -wv for v, wV. Formally, Λ(V) is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative K-algebra containing V with alternating multiplication on V must contain a homomorphic image of Λ(V). In other words, the exterior algebra has the following universal property:[4] In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, associativity is a property that a binary operation can have. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

Given any unital associative K-algebra A and any K-linear map j : VA such that j(v)j(v) = 0 for every v in V, then there exists precisely one unital algebra homomorphism f : Λ(V) → A such that f(v) = j(v) for all v in V. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... A homomorphism between two algebras over a field K, A and B, is a map such that for all k in K and x,y in A, F(kx)=kF(x) F(x+y)=F(x)+F(y) F(xy)=F(x)F(y) Categories: Math stubs | Algebra ...

To construct the most general algebra that contains V and whose multiplication is alternating on V, it is natural to start with the most general algebra that contains V, the tensor algebra T(V), and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided ideal I in T(V) generated by all elements of the form vv for v in V, and define Λ(V) as the quotient Image File history File links ExteriorAlgebra-01. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

Λ(V) = T(V)/I

(and use Λ as the symbol for multiplication in Λ(V)). It is then straightforward to show that Λ(V) contains V and satisfies the above universal property.

As a consequence of this construction, the operation of assigning to a vector space V its exterior algebra Λ(V) is a functor from the category of vector spaces to the category of algebras. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ...

Rather than defining Λ(V) first and then identifying the exterior powers Λk(V) as certain subspaces, one may alternatively define the spaces Λk(V) first and then combine them to form the algebra Λ(V). This approach is often used in differential geometry and is described in the next section.

### Generalizations

Given a commutative ring R and an R-module M, we can define the exterior algebra Λ(M) just as above, as a suitable quotient of the tensor algebra T(M). It will satisfy the analogous universal property. Many of the properties of Λ(M) also require that M be a projective module. Where finite-dimensionality is used, the properties further require that M be finitely generated and projective. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

## Duality

### Alternating operators

Given two vector spaces V and X, an alternating operator (or anti-symmetric operator) from Vk to X is a multilinear map In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable. ...

f: VkX

such that whenever v1,...,vk are linearly dependent vectors in V, then In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

f(v1,...,vk) = 0.

The most famous example is the determinant, an alternating operator from (Kn)n to K. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

The map

w: Vk → Λk(V)

which associates to k vectors from V their wedge product, i.e. their corresponding k-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on Vk: given any other alternating operator f : VkX, there exists a unique linear map φ: Λk(V) → X with f = φ o w. This universal property characterizes the space Λk(V) and can serve as its definition. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

### Alternating multilinear forms

The above discussion specializes to the case when X = K, the base field. In this case an alternating multilinear function

f : VkK

is called an alternating multilinear form. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the multiplication of such a map with a scalar, is again alternating. If V has finite dimension n, then this space can be identified with Λk(V), where V denotes the dual space of V. In particular, the dimension of the space of anti-symmetric maps from Vk to K is the binomial coefficient n choose k. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...

Under this identification, and if the base field is R or C, the wedge product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : VkK and η : VmK are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their wedge product is the sum of the numbers of their variables. It is defined as follows: In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...

$omegawedgeeta=frac{(k+m)!}{k!,m!}{rm Alt}(omegaotimeseta)$

where the alternation Alt of a multilinear map is defined to be the signed average of the values over all the permutations of its variables: Permutation is the rearrangement of objects or symbols into distinguishable sequences. ...

${rm Alt}(omega)(x_1,ldots,x_k)=frac{1}{k!}sum_{sigmain S_k}{rm sgn}(sigma),omega(x_{sigma(1)},ldots,x_{sigma(k)})$

This definition of the wedge product is well-defined even if the fields K has finite characteristic, if one considers an equivalent version of the above that does not use factorials or any constants: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ...

$omega wedge eta(x_1,ldots,x_{k+m}) = sum_{sigma in Sh_{k,m}} {rm sgn}(sigma),omega(x_{sigma(1)}, ldots, x_{sigma(k)}) eta(x_{sigma(k+1)}, ldots, x_{sigma(k+m)})$,

where here $Sh_{k,m} subset S_{k+m}$ is the subset of k,m shuffles: permutations σ sending $1,2,ldots,k$ to numbers $sigma(1) < sigma(2) < cdots < sigma(k)$, and $k+1,k+2,ldots,k+m$ to numbers $sigma(k+1). Let and be positive natural numbers. ...

(Note. Some conventions, particularly in physics, define the wedge product as

$omegawedgeeta={rm Alt}(omegaotimeseta).$

This convention is not adopted here, but see the Alternating tensor algebra section below for further details.)

### Bialgebra structure

In formal terms, there is a correspondence between the graded dual of the graded algebra Λ(V) and alternating multilinear forms on V. The wedge product of multilinear forms defined above is dual to a coproduct defined on Λ(V), giving the structure of a coalgebra. In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In mathematics, coalgebras are structures that are in a certain sense dual to the unital associative algebras. ...

The coproduct is a linear function Δ : Λ(V) → Λ(V) ⊗ Λ(V) given on decomposable elements by

$Delta(x_1wedgedotswedge x_k) = sum_{p=0}^k sum_{sigmain Sh_{p,k-p}} {rm sgn}(sigma) (x_{sigma(1)}wedgedotswedge x_{sigma(p)})otimes (x_{sigma(p+1)}wedgedotswedge x_{sigma(k)}).$

This extends by linearity to an operation defined on the whole exterior algebra. In terms of the coproduct, the wedge product on the dual space is just the graded dual of the coproduct:

$(alphawedgebeta)(x_1wedgedotswedge x_k) = (alphaotimesbeta)left(Delta(x_1wedgedotswedge x_k)right)$

where 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 o α⊗β o Δ, where ε is the counit, as defined presently).

The counit is the homomorphism ε : Λ(V) → K which returns the 0-graded component of its argument. The coproduct and counit, along with the wedge product, define the structure of a bialgebra on the exterior algebra. In mathematics, a bialgebra over a field K is a structure which is both a unital associative algebra and a coalgebra over K, such that the comultiplication and the counit are both unital algebra homomorphisms. ...

### The interior product

See also: interior product

Suppose that V is finite-dimensional. If V* denotes the dual space to the vector space V, then for each α ∈ V*, it is possible to define an antiderivation on the algebra Λ(V), In mathematics, the interior product is a degree âˆ’1 derivation on the exterior algebra of differential forms on a smooth manifold. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A &#8594; A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

$i_alpha:Lambda^k VrightarrowLambda^{k-1}V.$

This derivation is called the interior product with α, or sometimes the insertion operator.

Suppose that w ∈ ΛkV. Then w is a multilinear mapping of V* to R, so it is defined by its values on the k-fold Cartesian product V*× V*× ... × V*. If u1, u2, ..., uk-1 are k-1 elements of V*, then define In mathematics, the Cartesian product is a direct product of sets. ...

$(i_alpha {bold w})(u_1,u_2dots,u_{k-1})={bold w}(alpha,u_1,u_2,dots, u_{k-1})$

Additionally, let iαf = 0 whenever f is a pure scalar (i.e., belonging to Λ0V).

#### Axiomatic characterization and properties

The interior product satisfies the following properties:

1. For each k and each α ∈ V*,
$i_alpha:Lambda^kVrightarrow Lambda^{k-1}V.$
(By convention, Λ-1 = 0.)
2. If v is an element of V ( = Λ1V), then iαv = α(v) is the dual pairing between elements of V and elements of V*.
3. For each α ∈ V*, iα is a graded derivation of degree -1:
$i_alpha (awedge b) = (i_alpha a)wedge b + (-1)^{deg a}awedge (i_alpha b)$.

In fact, these three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case. In abstract algebra, a derivation on an algebra A over a ring or a field k is a linear map D : A â†’ A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

Further properties of the interior product include:

• $i_alphacirc i_alpha = 0.$
• $i_alphacirc i_beta = -i_betacirc i_alpha.$

### Hodge duality

Main article: Hodge dual

Suppose that V has finite dimension n. Then the interior product induces a canonical isomorphism of vector spaces In mathematics, the Hodge star operator or Hodge dual is a signficant linear map introduced in general by W. V. D. Hodge. ...

$Lambda^k(V^*) otimes Lambda^n(V) to Lambda^{n-k}(V).$

In the geometrical setting, a non-zero element of the top exterior power Λn(V) (which is a one-dimensional vector space) is sometimes called a volume form (or orientation form, although this term may sometimes lead to ambiguity.) Relative to a given volume form σ, the isomorphism is given explicitly by In mathematics, a volume form is a nowhere zero differential n-form on an oriented n-manifold. ...

$alpha in Lambda^k(V^*) mapsto i_alphasigma in Lambda^{n-k}(V).$

If, in addition to a volume form, the vector space V is equipped with an inner product identifying V with V*, then the resulting isomorphism is called the Hodge dual (or more commonly the Hodge star operator) In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

$* : Lambda^k(V) rightarrow Lambda^{n-k}(V).$

The composite of * with itself maps Λk(V) → Λk(V) and is always a scalar multiple of the identity map. In most applications, the volume form is compatible with the inner product in the sense that it is a wedge product of an orthonormal basis of V. In this case, In mathematics, an orthonormal basis of an inner product space V(i. ...

$*circ * : Lambda^k(V) to Lambda^k(V) = (-1)^{k(n-k) + q}I$

where I is the identity, and the inner product has metric signature (p,q) — p plusses and q minuses. The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...

Along with the bialgebra structure, the Hodge star operator on Λ(V) defines the antipode map for a Hopf algebra on the exterior algebra. In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Î” is the comultiplication of the bialgebra, âˆ‡ its multiplication, Î· its unit and Îµ its counit. ...

## Functoriality

Suppose that V and W are a pair of vector spaces and f : VW is a linear transformation. Then, by the universal construction, there exists a unique homomorphism of graded algebras In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

$Lambda(f) : Lambda(V)rightarrow Lambda(W)$

such that

$Lambda(f)|_{Lambda^1(V)} = f : V=Lambda^1(V)rightarrow W=Lambda^1(W).$

In particular, Λ(f) preserves homogeneous degree. The k-graded components of Λ(f) are given on decomposable elements by

$Lambda(f)(x_1wedge dots wedge x_k) = f(x_1)wedgedotswedge f(x_k).$

Let

$Lambda^k(f) = Lambda(f)_{Lambda^k(V)} : Lambda^k(V) rightarrow Lambda^k(W).$

The components of the transformation Λ(k) relative to a basis of V and W is the matrix of k × k minors of f. In particular, if V = W and V is of finite dimension n, then Λn(f) is a mapping of a one-dimensional vector space Λn to itself, and is therefore given by a scalar: the determinant of f. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

### Exactness

The functor Λ is exact, meaning that if In homological algebra, an exact functor is one which preserves exact sequences. ...

$0rightarrow Urightarrow Vrightarrow Wrightarrow 0$

is a short exact sequence of vector spaces, then In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

$0rightarrow Lambda(U)rightarrow Lambda(V)rightarrow Lambda(W)rightarrow 0$

is also exact.[5]

One consequence of the exactness is that the exterior powers of a direct sum of two vector spaces decompose into tensor products: In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

$Lambda^k(Voplus W)= bigoplus_{a+b=k}Lambda^a(V)otimesLambda^b(W).$

In particular, if

$0to Uto Vto Wto 0$

is a short exact sequence of vector spaces, with dim(U) = a, dim(V) = a+b, and dim(W) = b, then

$Lambda^{a+b}(V)=Lambda^a(U)otimesLambda^b(W).$

## The alternating tensor algebra

If K is a field of characteristic 0,[6] then the exterior algebra of a vector space V can be canonically identified with the vector subspace of T(V) consisting of antisymmetric tensors. Recall that the exterior algebra is the quotient of T(V) by the ideal I generated by xx. In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. ...

Let Tr(V) be the space of homogeneous tensors of rank r. This is spanned by decomposable tensors

$v_1otimesdotsotimes v_r,quad v_iin V.$

The antisymmetrization (or sometimes the skew-symmetrization) of a decomposable tensor is defined by

$text{Alt}(v_1otimesdotsotimes v_r) = frac{1}{r!}sum_{sigmainmathfrak{S}_r} {rm sgn}(sigma) v_{sigma(1)}otimesdotsotimes v_{sigma(r)}$

where the sum is taken over the symmetric group of permutations on the symbols {1,...,r}. This extends by linearity and homogeneity to an operation, also denoted by Alt, on the full tensor algebra T(V). The image Alt(T(V)) is the alternating tensor algebra, denoted A(V). This is a vector subspace of T(V), and it inherits the structure of a graded vector space from that on T(V). It carries an associative graded product $widehat{otimes}$ defined by In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

$t widehat{otimes} s = text{Alt}(totimes s).$

Although this product differs from the tensor product, the kernel of Alt is precisely the ideal I (again, assuming that K has characteristic 0), and there is a canonical isomorphism

$A(V)cong Lambda(V).$

### Index notation

Suppose that V has finite dimension n, and that a basis e1, ..., en of V is given. then any alternating tensor t ∈ Ar(V) ⊂ Tr(V) can be written in index notation as Index notation is used in mathematics to refer to the elements of matrices or the components of a vector. ...

$t = t^{i_1i_2dots i_r}, {mathbf e}_{i_1}otimes {mathbf e}_{i_2}otimesdotsotimes {mathbf e}_{i_r}$

where ti1...ir is completely antisymmetric in its indices. In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. ...

The wedge product of two alternating tensors t and s of ranks r and p is given by

$twidehat{otimes} s = frac{1}{(r+p)!}sum_{sigmain {mathfrak S}_{r+p}}text{sgn}(sigma)t^{i_{sigma(1)}dots i_{sigma(r)}}s^{i_{sigma(r+1)}dots i_{sigma(r+p)}} {mathbf e}_{i_1}otimes {mathbf e}_{i_2}otimesdotsotimes {mathbf e}_{i_{r+p}}.$

The components of this tensor are precisely the skew part of the components of the tensor product st, denoted by square brackets on the indices:

$(twidehat{otimes} s)^{i_1dots i_{r+p}} = t^{[i_1dots i_r}s^{i_{r+1}dots i_{r+p}]}$

The interior product may also be described in index notation as follows. Let $t = t^{i_0i_1dots i_{r-1}}$ be an antisymmetric tensor of rank r. Then, for α ∈ V*, iαt is an alternating tensor of rank r-1, given by

$(i_alpha t)^{i_1dots i_{r-1}}=rsum_{j=0}^nalpha_j t^{ji_1dots i_{r-1}}$.

where n is the dimension of V.

## Applications

### Linear geometry

The decomposable k-vectors have geometric interpretations: the bivector $uwedge v$ represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented parallelogram with sides u and v. Analogously, the 3-vector $uwedge vwedge w$ represents the spanned 3-space weighted by the volume of the oriented parallelepiped with edges u, v, and w. This article or section is not written in the formal tone expected of an encyclopedia article. ... In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek Ï€Î±ÏÎ±Î»Î»Î·Î»-ÎµÏ€Î¯Ï€ÎµÎ´Î¿Î½, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...

### Differential geometry

The exterior algebra has notable applications in differential geometry, where it are used to define differential forms. A differential forms can intuitively be interpreted as a function on weighted subspaces of the tangent space of a differentiable manifold. As a consequence, there is a natural wedge product for differential forms. Differential forms play a major role in diverse areas of differential geometry. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... This article or section is in need of attention from an expert on the subject. ...

### Representation theory

In representation theory, the exterior algebra is one of the two fundamental Schur functors on the category of vector spaces, the other being the symmetric algebra. Together, these constructions are used to generate the irreducible representations of the general linear group. In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that the image of the element corresponds to an irreducible representation of the symmetric group over the complex numbers. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the term irreducible is used in several ways. ... In mathematics, the general linear group of degree n is the set of nÃ—n invertible matrices, together with the operation of ordinary matrix multiplication. ...

### Physics

The exterior algebra is an archetypal example of a superalgebra, which plays a fundamental role in physical theories pertaining to fermions and supersymmetry. For a physical discussion, see Grassmann number. For various other applications of related ideas to physics, see superspace and supergroup (physics). In mathematics and theoretical physics, a superalgebra over a field K generally refers to a Z2-graded algebra over K (here Z2 is the cyclic group of order 2). ... In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ... This article or section is in need of attention from an expert on the subject. ... In mathematical physics, a Grassmann number (also called an anticommuting number) is a quantity that anticommutes with other Grassmann numbers but commutes with ordinary numbers , In particular, the square of any Grassmann number vanishes: The integration of Grassmannvariables needs to fullfill the following properties * linearity * partial integrationsformula This results in... Superspace has had two meanings in physics. ... The concept of supergroup is a generalization of a that of group. ...

## History

The exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension.[7] This 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. Hermann GÃ¼nther Grassmann (April 15, 1809, Stettin â€“ September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors. It was thus a calculus, much like the propositional calculus, except focused exclusively on the task of formal reasoning in geometrical terms.[8] In 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. A multivector is an element of a geometric algebra , most generally a summation of elements of different or mixed grade such as the summation of a scalar, a vector, and a bivector although a few authors favour polyvector for mixed grade elements, reserving multivector for geoemetric algebra elements consisting of... In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ...

The import of this new theory of vectors and multivectors was lost to mid 19th century mathematicians,[9] until being thoroughly vetted by Giuseppe Peano in 1888. Peano'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, Elie Cartan, and Gaston Darboux) who applied Grassmann's ideas to the calculus of differential forms. Giuseppe Peano Giuseppe Peano (August 27, 1858 â€“ April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... Jules TuPac Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... Jean Gaston Darboux (August 14, 1842, Nîmes &#8211; February 23, 1917, Paris) was a French mathematician. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

A short while later, Alfred North Whitehead, borrowing from the ideas of Peano and Grassmann, introduced his universal algebra. This then paved the way for the 20th century developments of abstract algebra by placing the axiomatic notion of an algebraic system on a firm logical footing. Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England â€“ December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ... Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...

## Notes

1. ^ Grassmann (1844) introduced these as extended algebras (cf. Clifford, 1878). He used the word äußere (literally translated as outer, or exterior) only to indicate the produkt he defined, which is nowadays conventionally called exterior product, probably to distinguish it from the outer product as defined in modern linear algebra.
2. ^ Note that these three properties are only valid for the vectors in V, not for all elements of the algebra Λ(V). The defining property (1) and property (3) are equivalent; properties (1) and (2) are equivalent unless the characteristic of K is two.
3. ^ This axiomatization of areas is due to Leopold Kronecker and Karl Weierstrass (see Bourbaki (1989), Historical Note). For a modern treatment, see MacLane and Birkhoff (1999) Theorem IX.2.2. For an elementary treatment, see Strang (1993), Chapter 5.
4. ^ See Bourbaki (1989) III.7.1, and MacLane and Birkhoff (1999) Theorem XVI.6.8. More detail on universal properties in general can be found in MacLane and Birkhoff (1999) Chapter VI, and throughout the works of Bourbaki.
5. ^ This also holds in greater generality if U, V, and W are projective modules over a commutative ring. In the non-projective case, Λ is only right-exact.
6. ^ See Bourbaki (1989) III.7.5 for generalizations.
7. ^ Kannenberg (2000) published a translation of Grassmann's work in English; he translated Ausdehnungslehre as Extension Theory.
8. ^ Authors have in the past referred to this calculus variously as the calculus of extension (Whitehead, 1898; Forder, 1941), or extensive algebra (Clifford, 1878), and recently as extended vector algebra (Browne, 2007), not to be confused with the modern notion of algebra over a field.
9. ^ Bourbaki, Algebra (1989) p. 661.

Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... Whitehead can refer to: People: Alfred North Whitehead, a British philosopher and mathematician Cortlandt Whitehead (1842-), bishop Commander Edward Whitehead (1908-1978), spokesperson and later president of Schweppes [1] Ennis Clement Whitehead (1895-1964), Lieutenant General U.S. Air Force [2] Gary Whitehead, American poet Gustave Whitehead, German-American aviation... Clifford is both a given name and a surname that applies to a number of individuals or places. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...

## References

### Mathematical references

• Bishop, R. and Goldberg, S.I. (1980). Tensor analysis on manifolds. Dover. ISBN 0-486-64039-6.
Includes a treatment of alternating tensors and alternating forms, as well as a detailed discussion of Hodge duality from the perspective adopted in this article.
• Bourbaki, Nicolas (1989). Elements of mathematics, Algebra I. Springer-Verlag. ISBN 3-540-64243-9.
This is the main mathematical reference for the article. It introduces the exterior algebra of a module over a commutative ring (although this article specializes primarily to the case when the ring is a field), including a discussion of the universal property, functoriality, duality, and the bialgebra structure. See chapters III.7 and III.11.
Chapter XVI sections 6-10 give a more elementary account of the exterior algebra, including duality, determinants and minors, and alternating forms.
Contains a classical treatment of the exterior algebra as alternating tensors, and applications to differential geometry.

Nicolas Bourbaki is the collective allonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ... Saunders Mac Lane (born 4 August 1909) is a US mathematician. ... Garrett Birkhoff (January 19, 1911, Princeton, New Jersey, USA - November 22, 1996, Water Mill, New York, USA) was an American mathematician. ... Shlomo Zvi Sternberg is a leading mathematician, known for his work in geometry, particularly symplectic geometry. ...

### Historical references

• Bourbaki, Nicolas (1989). "Historical note on chapters II and III", Elements of mathematics, Algebra I. Springer-Verlag.
• Clifford, W. (1878). "Applications of Grassmann's Extensive Algebra". American Journal of Mathematics 1 (4): 350-358.
• Forder, H. G. (1941). The Calculus of Extension. Cambridge University Press.
• Grassmann, Hermann (1844). Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik.  (The Linear Extension Theory - A new Branch of Mathematics)
• Grassmann, Hermann (2000). Extension Theory (translation by L.C. Kannenberg). American Mathematical Society. ISBN 0821820311.
• Peano, Giuseppe (1888). Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva.  [Geometric Calculus according to Grassmann's Ausdehnungslehre, preceded by the Operations of Deductive Logic]
• Whitehead, Alfred North (1898). A Treatise on Universal Algebra, with Applications. Cambridge.

### Other references and further reading

• Browne, J.M. (2007). Grassmann algebra - Exploring applications of Extended Vector Algebra with Mathematica. Published on line.
An introduction to the exterior algebra, and geometric algebra, with a focus on applications. Also includes a history section and bibliography.
• Strang, G. (1993). Introduction to linear algebra. Wellesley-Cambridge Press.
Includes an elementary treatment of the axiomatization of determinants as signed areas, volumes, and higher-dimensional volumes.
• Onishchik, A.L., Exterior algebra, SpringerLink Encyclopaedia of Mathematics (2001)

A geometric algebra is a multilinear algebra with a geometric interpretation. ... Gilbert Strang is an American mathematician who published (with George Fix) An Analysis of The Finite Element Method in 1973. ...

## See also

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