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In mathematics, the exterior algebra, or Grassmann algebra, named after
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
, is an algebra that uses the exterior product or wedge product as its multiplication. In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the exterior product or wedge product of vectors is an algebraic construction used in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
to study
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
s,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
s, and their higher-dimensional analogues. The exterior product of two vectors u and  v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and  v, which in three dimensions can also be computed using the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is
anticommutative In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswappe ...
, meaning that u \wedge v = -(v \wedge u) for all vectors u and  v, but, unlike the cross product, the exterior product is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
. When regarded in this manner, the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number ''k'' of vectors can be defined and is sometimes called a ''k''-blade. It lives in a space known as the ''k-''th exterior power. The magnitude of the resulting ''k''-blade is the oriented hypervolume of the ''k''-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not only ''k''-blades, but sums of ''k''-blades; such a sum is called a ''k''-vector. The ''k''-blades, because they are simple products of vectors, are called the simple elements of the algebra. The ''rank'' of any ''k''-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
, which means that \alpha \wedge (\beta \wedge \gamma) = (\alpha \wedge \beta) \wedge \gamma for any elements \alpha, \beta, \gamma. The ''k''-vectors have degree ''k'', meaning that they are sums of products of ''k'' vectors. When elements of different degrees are multiplied, the degrees add like multiplication of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s. This means that the exterior algebra is a
graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
. The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, and for other structures of interest in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms, and the pairing between the exterior algebra and its dual is given by the
interior product In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of ...
.


Motivating examples

The first two examples assume a metric tensor field and an orientation; the third example does not assume either.


Areas in the plane

The Cartesian plane \mathbb R^2 is a real vector space equipped with a basis consisting of a pair of
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
s : _1 = \begin1\\0\end,\quad _2 = \begin0\\1\end, with the orientation \mathbf e_1 \times \mathbf e_2 and with the metric \begin1 & 0\\0 & 1\end. Suppose that :\mathbf = \begina\\b\end = a \mathbf_1 + b \mathbf_2, \quad \mathbf = \beginc\\d\end = c \mathbf_1 + d \mathbf_2 are a pair of given vectors in \R^2, written in components. There is a unique parallelogram having v and w as two of its sides. The ''area'' of this parallelogram is given by the standard
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
formula: : \text = \Bigl, \det \begin \mathbf & \mathbf \end \Bigr, = \Biggl, \det \begin a & c \\ b & d \end \Biggr, = \left, ad - bc \ . Consider now the exterior product of v and w: :\begin \mathbf \wedge \mathbf &= (a\mathbf_1 + b\mathbf_2) \wedge (c\mathbf_1 + d\mathbf_2) \\ &= ac\mathbf_1 \wedge \mathbf_1 + ad\mathbf_1 \wedge \mathbf_2 + bc\mathbf_2 \wedge \mathbf_1 + bd\mathbf_2 \wedge \mathbf_2 \\ &= \left( ad - bc \right)\mathbf_1 \wedge \mathbf_2 \end where the first step uses the distributive law for the exterior product, and the last uses the fact that the exterior product is alternating, and in particular \mathbf_2 \wedge \mathbf_1 = -(\mathbf_1 \wedge \mathbf_2). (The fact that the exterior product is alternating also forces \mathbf_1 \wedge \mathbf_1 = \mathbf_2 \wedge \mathbf_2 = 0.) Note that the coefficient in this last expression is precisely the determinant of the matrix . 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 In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
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 denotes the signed area of the parallelogram of which the pair of vectors v and w form two adjacent sides, then A must satisfy the following properties: # for any real numbers ''r'' and ''s'', 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). # , since the area of the degenerate parallelogram determined by v (i.e., a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
) is zero. # , since interchanging the roles of v and w reverses the orientation of the parallelogram. # for any real number ''r'', since adding a multiple of w to v affects neither the base nor the height of the parallelogram and consequently preserves its area. # , since the area of the unit square is one. With the exception of the last property, the exterior product of two vectors satisfies the same properties as the area. In a certain sense, the exterior product generalizes the final property by allowing the area of a parallelogram to be compared to that of any chosen parallelogram in a parallel plane (here, the one with sides e1 and e2). In other words, the exterior product provides a ''basis-independent'' formulation of area.


Cross and triple products

For vectors in a 3-
dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
oriented
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
with a bilinear scalar product, the exterior algebra is closely related to the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
and
triple product In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
. Using a
standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
, the exterior product of a pair of vectors : \mathbf = u_1 \mathbf_1 + u_2 \mathbf_2 + u_3 \mathbf_3 and : \mathbf = v_1 \mathbf_1 + v_2 \mathbf_2 + v_3 \mathbf_3 is : \mathbf \wedge \mathbf = (u_1 v_2 - u_2 v_1) (\mathbf_1 \wedge \mathbf_2) + (u_2 v_3 - u_3 v_2) (\mathbf_2 \wedge \mathbf_3) + (u_3 v_1 - u_1 v_3) (\mathbf_3 \wedge \mathbf_1) , where is a basis for the three-dimensional space \bigwedge\nolimits^2\left(\mathbb R^3\right). The coefficients above are the same as those in the usual definition of the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of vectors in three dimensions with a given orientation, the only differences being that the exterior product is not an ordinary vector, but instead is a
2-vector In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors ( ...
, and that the exterior product does not depend on the choice of orientation . Bringing in a third vector : \mathbf = w_1 \mathbf_1 + w_2 \mathbf_2 + w_3 \mathbf_3, the exterior product of three vectors is : \mathbf \wedge \mathbf \wedge \mathbf = (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_1 \wedge \mathbf_2 \wedge \mathbf_3) where is the basis vector for the one-dimensional space \bigwedge\nolimits^3\left(\mathbb R^3\right). The scalar coefficient is the
triple product In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
of the three vectors. The cross product and triple product in a three dimensional Euclidean vector space each admit both geometric and algebraic interpretations, via Hodge star duality. The cross product 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 a signed scalar representing a geometric oriented 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: it, too, can be identified with oriented lines, areas, volumes, etc., that are spanned by one, two or more vectors. The exterior product generalizes these geometric notions to all vector spaces and to any number of dimensions, even in the absence of a scalar product.


Electromagnetic field

In Einstein's theories of relativity, the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
is generally given as a differential 2-form F = dA in
4-space A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
or as the equivalent alternating tensor field F_ = A_ = A_, the
Electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. ...
. Then dF = ddA = 0 or the equivalent Bianchi identity F_ = F_ = 0. None of this requires a metric. Adding the Lorentz metric and an orientation provides the Hodge star operator \star and thus makes it possible to define J = dF or the equivalent tensor
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
J^i = F^_ = F^_ where F^ = g^g^F_.


Formal definitions and algebraic properties

The exterior algebra \bigwedge(V) of a vector space over a field is defined as the quotient algebra of the tensor algebra by the two-sided
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
generated by all elements of the form for (i.e. all tensors that can be expressed as the tensor product of a vector in by itself). The ideal ''I'' contains the ideal ''J'' generated by elements of the form x \otimes y + y \otimes x = (x + y) \otimes (x + y) - x \otimes x - y \otimes y and these ideals coincide if \operatorname(K) \ne 2 (if \operatorname(K) = 2, these ideals are different except for the zero vector space). So, : (V) = T(V)/I is an
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
. Its multiplication is called the ''exterior product'', and denoted . This means that the product of \bigwedge(V) is induced by the tensor product of . As , , and \left(T^0(V) \oplus T^1(V)\right) \cap I = \, the inclusions of and in induce injections of and into \bigwedge(V). These injections are commonly considered as inclusions, and called ''natural embeddings'', ''natural injections'' or ''natural inclusions''. The word ''canonical'' is also commonly used in place of ''natural''.


Alternating product

The exterior product is by construction ''alternating'' on elements of V, which means that x \wedge x = 0 for all x \in V, by the above construction. It follows that the product is also
anticommutative In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswappe ...
on elements of V, for supposing that x, y \in V, : 0 = (x + y) \wedge (x + y) = x \wedge x + x \wedge y + y \wedge x + y \wedge y = x \wedge y + y \wedge x hence : x \wedge y = -(y \wedge x). More generally, if ''σ'' is a
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
of the integers , and ''x''1, ''x''2, ..., ''x''''k'' are elements of ''V'', it follows that : x_ \wedge x_ \wedge \cdots \wedge x_ = \operatorname(\sigma)x_1 \wedge x_2 \wedge \cdots \wedge x_k, where sgn(''σ'') is the signature of the permutation ''σ''. In particular, if ''x''''i'' = ''x''''j'' for some , then the following generalization of the alternating property also holds: : x_ \wedge x_ \wedge \cdots \wedge x_ = 0. Together with the distributive property of the exterior product, one further generalization is that if and only if \ is a linearly dependent set of vectors, then : x_ \wedge x_ \wedge \cdots \wedge x_ = 0.


Exterior power

The ''k''th exterior power of ''V'', denoted \bigwedge\nolimits^k(V), is the vector subspace of \bigwedge(V) spanned by elements of the form : x_1 \wedge x_2 \wedge \cdots \wedge x_k,\quad x_i \in V, i=1,2,\ldots, k. If \alpha \in \bigwedge\nolimits^k(V), then ''α'' is said to be a ''k''-vector. If, furthermore, ''α'' can be expressed as an exterior product of ''k'' elements of ''V'', then ''α'' is said to be decomposable. Although decomposable ''k''-vectors span \bigwedge\nolimits^k(V), not every element of \bigwedge\nolimits^k(V) is decomposable. For example, in \mathbb R^4, the following 2-vector is not decomposable: : \alpha = e_1 \wedge e_2 + e_3 \wedge e_4. (This is a symplectic form, since .)


Basis and dimension

If the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of is and is a basis for , then the set : \ is a basis for \bigwedge\nolimits^k(V). The reason is the following: given any exterior product of the form : v_1 \wedge \cdots \wedge v_k , every vector can be written as a linear combination of the basis vectors ; using the bilinearity of the exterior product, this can be expanded to a linear combination of exterior products of those basis vectors. Any 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. In general, the resulting coefficients of the basis -vectors can be computed as the minors of the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
that describes the vectors in terms of the basis . By counting the basis elements, the dimension of \bigwedge\nolimits^k(V) is equal to a
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
: : \dim ^k(V) = \binom\,. where is the dimension of the ''vectors'', and is the number of vectors in the product. The binomial coefficient produces the correct result, even for exceptional cases; in particular, \bigwedge\nolimits^k(V) = \ for . Any element of the exterior algebra can be written as a sum of -vectors. Hence, as a vector space the exterior algebra is a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
: (V) = ^0(V) \oplus ^1(V) \oplus ^2(V) \oplus \cdots \oplus ^n(V) (where by convention \bigwedge\nolimits^0(V) = K, the field underlying , and \bigwedge\nolimits^1(V) = V ), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2.


Rank of a ''k''-vector

If \alpha \in \bigwedge\nolimits^k(V), then it is possible to express ''α'' as a linear combination of decomposable ''k''-vectors: : \alpha = \alpha^ + \alpha^ + \cdots + \alpha^ where each ''α''(''i'') is decomposable, say : \alpha^ = \alpha^_1 \wedge \cdots \wedge \alpha^_k,\quad i = 1,2,\ldots, s. The rank of the ''k''-vector ''α'' is the minimal number of decomposable ''k''-vectors in such an expansion of ''α''. This is similar to the notion of tensor rank. Rank is particularly important in the study of 2-vectors . The rank of a 2-vector ''α'' can be identified with half the rank of the matrix of coefficients of ''α'' in a basis. Thus if ''e''''i'' is a basis for ''V'', then ''α'' can be expressed uniquely as : \alpha = \sum_a_e_i \wedge e_j where (the matrix of coefficients is skew-symmetric). The rank of the matrix ''a''''ij'' is therefore even, and is twice the rank of the form ''α''. In characteristic 0, the 2-vector ''α'' has rank ''p'' if and only if : \underset \neq 0 \ and \ \underset = 0.


Graded structure

The exterior product of a ''k''-vector with a ''p''-vector is a -vector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section : (V) = ^(V) \oplus ^(V) \oplus ^(V) \oplus \cdots \oplus ^(V) gives the exterior algebra the additional structure of a
graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
, that is : ^k(V) \wedge ^p(V) \sub ^(V). Moreover, if is the base field, we have : ^(V) = K and ^(V) = V. The exterior product is graded anticommutative, meaning that if \alpha \in \bigwedge\nolimits^k(V) and \beta \in \bigwedge\nolimits^p(V), then : \alpha \wedge \beta = (-1)^\beta \wedge \alpha. In 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 (a module that already carries its own gradation).


Universal property

Let be a vector space over the field . Informally, multiplication in \bigwedge(V) is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity v \wedge v = 0 for . Formally, \bigwedge(V) is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative -algebra containing with alternating multiplication on must contain a homomorphic image of \bigwedge(V). In other words, the exterior algebra has the following
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
:
Given any unital associative -algebra and any -
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
j : V \to A such that j(v)j(v) = 0 for every in , then there exists ''precisely one'' unital
algebra homomorphism In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF( ...
f : \bigwedge(V)\to A such that for all in (here is the natural inclusion of in \bigwedge(V), see above).
To construct the most general algebra that contains and whose multiplication is alternating on , it is natural to start with the most general associative algebra that contains , the tensor algebra , and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
in generated by all elements of the form for in , and define \bigwedge(V) as the quotient : (V) = T(V)/I\ (and use as the symbol for multiplication in \bigwedge(V) ). It is then straightforward to show that \bigwedge(V) contains and satisfies the above universal property. As a consequence of this construction, the operation of assigning to a vector space its exterior algebra \bigwedge(V) is a functor from the category of vector spaces to the category of algebras. Rather than defining \bigwedge(V) first and then identifying the exterior powers \bigwedge\nolimits^k(V) as certain subspaces, one may alternatively define the spaces \bigwedge\nolimits^k(V) first and then combine them to form the algebra \bigwedge(V). This approach is often used in differential geometry and is described in the next section.


Generalizations

Given a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''R'' and an ''R''-
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
''M'', we can define the exterior algebra \bigwedge(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 \bigwedge(M) also require that ''M'' be a
projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characteriz ...
. Where finite dimensionality is used, the properties further require that ''M'' be finitely generated and projective. Generalizations to the most common situations can be found in . Exterior algebras of
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s are frequently considered in geometry and topology. There 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. More general exterior algebras can be defined for sheaves of modules.


Alternating tensor algebra

If ''K'' is a field of characteristic 0, then the exterior algebra of a vector space ''V'' over ''K'' 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 elements of the form . Let T''r''(''V'') be the space of homogeneous tensors of degree ''r''. This is spanned by decomposable tensors : v_1 \otimes \cdots \otimes v_r,\quad v_i \in V. The antisymmetrization (or sometimes the skew-symmetrization) of a decomposable tensor is defined by : \operatorname(v_1 \otimes \cdots \otimes v_r) = \frac\sum_ \operatorname(\sigma) v_ \otimes \cdots \otimes v_ where the sum is taken over the symmetric group of permutations on the symbols 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 defined by : t~\widehat~s = \operatorname(t \otimes 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 (V).


Index notation

Suppose that ''V'' has finite dimension ''n'', and that a basis of ''V'' is given. Then any alternating tensor can be written in
index notation In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to t ...
as : t = t^\, _ \otimes _ \otimes \cdots \otimes _, where ''t''''i''1⋅⋅⋅''i''''r'' is completely antisymmetric in its indices. The exterior product of two alternating tensors ''t'' and ''s'' of ranks ''r'' and ''p'' is given by : t~\widehat~s = \frac\sum_\operatorname(\sigma)t^ s^ _ \otimes _ \otimes \cdots \otimes _. The components of this tensor are precisely the skew part of the components of the tensor product , denoted by square brackets on the indices: : (t~\widehat~s)^ = t^s^. The interior product may also be described in index notation as follows. Let t = t^ be an antisymmetric tensor of rank ''r''. Then, for , ''i''''α''''t'' is an alternating tensor of rank , given by : (i_\alpha t)^ = r\sum_^n\alpha_j t^. where ''n'' is the dimension of ''V''.


Duality


Alternating operators

Given two vector spaces ''V'' and ''X'' and a natural number ''k'', an alternating operator from ''V''''k'' to ''X'' is a multilinear map : f\colon V^k \to X such that whenever ''v''1, ..., ''v''''k'' are linearly dependent vectors in ''V'', then : f(v_1,\ldots, v_k) = 0. The map : w\colon V^k \to ^(V) which associates to k vectors from V their exterior product, i.e. their corresponding k -vector, is also alternating. In fact, this map is the "most general" alternating operator defined on V^k; given any other alternating operator f : V^k \rightarrow X, there exists a unique
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
\phi : \wedge^(V) \rightarrow X with f = \phi \circ w. This
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
characterizes the space \wedge^(V) and can serve as its definition.


Alternating multilinear forms

The above discussion specializes to the case when , the base field. In this case an alternating multilinear function : f : V^k \to K\ is called an alternating multilinear form. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating. By the universal property of the exterior power, the space of alternating forms of degree ''k'' on ''V'' is naturally isomorphic with the dual vector space \bigl(\bigwedge\nolimits^k(V)\bigr)^*. If ''V'' is finite-dimensional, then the latter is to \bigwedge\nolimits^k\left(V^*\right). In particular, if ''V'' is ''n''-dimensional, the dimension of the space of alternating maps from ''V''''k'' to ''K'' is the
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
\binom. Under such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose and are two anti-symmetric maps. As in the case of
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
s of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables. Depending on the choice of identification of elements of exterior power with multilinear forms, the exterior product is defined as : \omega \wedge \eta = \operatorname(\omega \otimes \eta) or as : \omega \wedge \eta = \frac\operatorname(\omega \otimes \eta), 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
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s of its variables: : \operatorname(\omega)(x_1,\ldots,x_k) = \frac\sum_\operatorname(\sigma)\, \omega(x_, \ldots, x_). When the field ''K'' has finite characteristic, an equivalent version of the second expression without any factorials or any constants is well-defined: : = \sum_ \operatorname(\sigma)\, \omega(x_, \ldots, x_)\, \eta(x_, \ldots, x_), where here is the subset of (''k'',''m'') shuffles:
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s ''σ'' of the set such that , and .


Interior product

Suppose that ''V'' is finite-dimensional. If ''V'' denotes the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
to the vector space ''V'', then for each , it is possible to define an
antiderivation In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Le ...
on the algebra \bigwedge(V), : i_\alpha:^k V \rightarrow ^V. This derivation is called the interior product with ''α'', or sometimes the insertion operator, or contraction by ''α''. Suppose that w \in \bigwedge\nolimits^k V. Then w is a multilinear mapping of ''V'' to ''K'', so it is defined by its values on the ''k''-fold
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
. If ''u''1, ''u''2, ..., ''u''''k''−1 are elements of ''V'', then define : (i_\alpha )(u_1,u_2,\ldots,u_) = (\alpha,u_1,u_2,\ldots, u_). Additionally, let whenever ''f'' is a pure scalar (i.e., belonging to \bigwedge\nolimits^0 V ).


Axiomatic characterization and properties

The interior product satisfies the following properties: # For each ''k'' and each , i_\alpha:^k V\rightarrow ^V. (By convention, \bigwedge\nolimits^V=\. ) # If ''v'' is an element of ''V'' ( = \bigwedge\nolimits^1 V ), then is the dual pairing between elements of ''V'' and elements of ''V''. # For each , ''i''''α'' is a graded derivation of degree −1: i_\alpha (a \wedge b) = (i_\alpha a) \wedge b + (-1)^a \wedge (i_\alpha b). These three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case. Further properties of the interior product include: * i_\alpha\circ i_\alpha = 0. * i_\alpha\circ i_\beta = -i_\beta\circ i_\alpha.


Hodge duality

Suppose that ''V'' has finite dimension ''n''. Then the interior product induces a canonical isomorphism of vector spaces : ^k(V^*) \otimes ^n(V) \to ^(V) by the recursive definition : i_ = i_\beta \circ i_\alpha. In the geometrical setting, a non-zero element of the top exterior power \bigwedge\nolimits^n(V) (which is a one-dimensional vector space) is sometimes called a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
(or orientation form, although this term may sometimes lead to ambiguity). The 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. Relative to the preferred volume form ''σ'', the isomorphism is given explicitly by : ^k(V^*) \to ^(V) : \alpha \mapsto i_\alpha\sigma . If, in addition to a volume form, the vector space ''V'' is equipped with an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
identifying ''V'' with ''V'', then the resulting isomorphism is called the Hodge star operator, which maps an element to its Hodge dual: : \star : ^k(V) \rightarrow ^(V) . The composition of \star with itself maps 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 an exterior product of an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
of ''V''. In this case, : \star \circ \star : ^k(V) \to ^k(V) = (-1)^\mathrm where id is the identity mapping, and the inner product has metric signature — ''p'' pluses and ''q'' minuses.


Inner product

For ''V'' a finite-dimensional space, an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
(or a pseudo-Euclidean inner product) on ''V'' defines an isomorphism of ''V'' with ''V'', and so also an isomorphism of \bigwedge\nolimits^k V with \bigl(\bigwedge\nolimits^k V\bigr)^*. The pairing between these two spaces also takes the form of an inner product. On decomposable ''k''-vectors, : \left\langle v_1 \wedge \cdots \wedge v_k, w_1 \wedge \cdots \wedge w_k\right\rangle = \det\bigl(\langle v_i,w_j\rangle\bigr), the determinant of the matrix of inner products. In the special case , the inner product is the square norm of the ''k''-vector, given by the determinant of the Gramian matrix . This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on \bigwedge\nolimits^k V. If ''e''''i'', , form an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
of ''V'', then the vectors of the form : e_ \wedge \cdots \wedge e_,\quad i_1 < \cdots < i_k, constitute an orthonormal basis for \bigwedge\nolimits^k(V) , a statement equivalent to the
Cauchy–Binet formula In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so t ...
. With respect to the inner product, exterior multiplication and the interior product are mutually adjoint. Specifically, for v \in \bigwedge\nolimits^(V), w \in \bigwedge\nolimits^(V), and x \in V, : \langle x \wedge \mathbf, \mathbf\rangle = \langle \mathbf, i_\mathbf\rangle where is the
musical isomorphism In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle \mathrmM and the cotangent bundle \mathrm^* M of a pseudo-Riemannian manifold induced b ...
, the linear functional defined by : x^\flat(y) = \langle x, y\rangle for all . This property completely characterizes the inner product on the exterior algebra. Indeed, more generally for v \in \bigwedge\nolimits^(V), w \in \bigwedge\nolimits^(V), and x \in \bigwedge\nolimits^(V), iteration of the above adjoint properties gives : \langle \mathbf \wedge \mathbf, \mathbf\rangle = \langle \mathbf, i_\mathbf\rangle where now x^\flat \in \bigwedge\nolimits^l\left(V^*\right) \simeq \bigl(\bigwedge\nolimits^l(V)\bigr)^* is the dual ''l''-vector defined by : \mathbf^\flat(\mathbf) = \langle \mathbf, \mathbf\rangle for all y \in \bigwedge\nolimits^l(V).


Bialgebra structure

There is a correspondence between the graded dual of the graded algebra \bigwedge(V) and alternating multilinear forms on ''V''. The exterior algebra (as well as the symmetric algebra) inherits a bialgebra structure, and, indeed, a
Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Swe ...
structure, from the tensor algebra. See the article on tensor algebras for a detailed treatment of the topic. The exterior product of multilinear forms defined above is dual to a coproduct defined on \bigwedge(V), giving the structure of a coalgebra. The coproduct is a linear function which is given by : \Delta(v) = 1 \otimes v + v \otimes 1 on elements ''v''∈''V''. The symbol 1 stands for the unit element of the field ''K''. Recall that so that the above really does lie in This definition of the coproduct is lifted to the full space \bigwedge(V) by (linear) homomorphism. The correct form of this homomorphism is not what one might naively write, but has to be the one carefully defined in the coalgebra article. In this case, one obtains : \Delta(v \wedge w) = 1 \otimes (v \wedge w) + v \otimes w - w \otimes v + (v \wedge w) \otimes 1. Expanding this out in detail, one obtains the following expression on decomposable elements: : \Delta(x_1 \wedge \cdots \wedge x_k) = \sum_^k \; \sum_ \; \operatorname(\sigma) (x_ \wedge \cdots \wedge x_) \otimes (x_ \wedge \cdots \wedge x_). where the second summation is taken over all -shuffles. The above is written with a notational trick, to keep track of the field element 1: the trick is to write x_0 = 1, and this is shuffled into various locations during the expansion of the sum over shuffles. The 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. Observe that the coproduct preserves the grading of the algebra. Extending to the full space \bigwedge(V), one has : \Delta:^k(V) \to \bigoplus_^k ^p(V) \otimes ^(V) The 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. Intuitively, 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 Any lingering doubt can be shaken by pondering the equalities and , which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols. This distinction is developed in greater detail in the article on tensor algebras. Here, there is much less of a problem, in that the alternating product ∧ clearly corresponds to multiplication in the bialgebra, leaving the symbol ⊗ free for use in the definition of the bialgebra. In practice, this presents no particular problem, as long as one avoids the fatal trap of replacing alternating sums of ⊗ by the wedge symbol, with one exception. One can construct an alternating product from ⊗, with the understanding that it works in a different space. Immediately below, an example is given: the alternating product for the ''dual space'' can be given in terms of the coproduct. The construction of the bialgebra here parallels the construction in the tensor algebra article almost exactly, except for the need to correctly track the alternating signs for the exterior algebra. In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct: : (\alpha \wedge \beta)(x_1 \wedge \cdots \wedge x_k) = (\alpha \otimes \beta)\left(\Delta(x_1 \wedge \cdots \wedge 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, , where ''ε'' is the counit, as defined presently). The counit is the homomorphism that returns the 0-graded component of its argument. The coproduct and counit, along with the exterior product, define the structure of a bialgebra on the exterior algebra. With an antipode defined on homogeneous elements by S(x) = (-1)^x, the exterior algebra is furthermore a
Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Swe ...
.


Functoriality

Suppose that ''V'' and ''W'' are a pair of vector spaces and is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
. Then, by the universal property, there exists a unique homomorphism of graded algebras : (f) : (V)\rightarrow (W) such that : (f)\left, _\right. = f : V=^1(V)\rightarrow W=^1(W). In particular, \bigwedge\left(f\right) preserves homogeneous degree. The ''k''-graded components of \bigwedge\left(f\right) are given on decomposable elements by : (f)(x_1 \wedge \cdots \wedge x_k) = f(x_1) \wedge \cdots \wedge f(x_k). Let : ^k(f) = (f)\left, _\right. : ^k(V) \rightarrow ^k(W). The components of the transformation \bigwedge\nolimits^k\left(f\right) relative to a basis of ''V'' and ''W'' is the matrix of minors of ''f''. In particular, if and ''V'' is of finite dimension ''n'', then \bigwedge\nolimits^n\left(f\right) is a mapping of a one-dimensional vector space \bigwedge\nolimits^n(V) to itself, and is therefore given by a scalar: the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of ''f''.


Exactness

If 0 \to U \to V \to W \to 0 is a short exact sequence of vector spaces, then : 0 \to ^1(U) \wedge (V) \to (V) \to (W) \to 0 is an exact sequence of graded vector spaces, as is : 0 \to \bigwedge(U) \to \bigwedge(V).


Direct sums

In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras: : (V \oplus W) \cong (V) \otimes (W). This is a graded isomorphism; i.e., : ^k(V \oplus W) \cong \bigoplus_ ^p(V) \otimes ^q(W). In greater generality, for a short exact sequence of vector spaces 0 \to U \mathrel V \mathrel W \to 0, there is a natural
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
: 0 = F^0 \subseteq F^1 \subseteq \cdots \subseteq F^k \subseteq F^ = ^k(V) where F^p for p \geq 1 is spanned by elements of the form u_1 \wedge \ldots \wedge u_ \wedge v_1 \wedge \ldots v_ for u_i \in U and v_i \in V. The corresponding quotients admit a natural isomorphism : F^/F^p \cong ^(U) \otimes ^p(W) given by u_1 \wedge \ldots \wedge u_ \wedge v_1 \wedge \ldots \wedge v_ \mapsto u_1 \wedge \ldots \wedge u_ \otimes g(v_1) \wedge \ldots \wedge g(v_). In particular, if ''U'' is 1-dimensional then : 0 \to U \otimes ^(W) \to ^k(V) \to ^k(W) \to 0 is exact, and if ''W'' is 1-dimensional then : 0 \to ^k(U) \to ^k(V) \to ^(U) \otimes W \to 0 is exact.


Applications


Linear algebra

In applications to
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, the exterior product provides an abstract algebraic manner for describing the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
and the minors of a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
. For 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). This suggests that the determinant can be ''defined'' in terms of the exterior product of the column vectors. Likewise, the minors of a matrix can be defined by looking at the exterior products of column vectors chosen ''k'' at a time. These ideas can be extended not just to matrices but to linear transformations as well: the determinant of a linear transformation is the factor by which it scales the oriented volume of any given reference parallelotope. So the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power. The action of a transformation on the lesser exterior powers gives a basis-independent way to talk about the minors of the transformation.


Technical details: Definitions

Let V be an ''n''-dimensional vector space over field K with basis \. * For A \in \operatorname(V), define \bigwedge^k A \in \operatorname\bigl(\bigwedge^k V\bigr) on simple tensors by ^k A(v_1 \wedge \cdots \wedge v_k) = Av_1 \wedge \cdots \wedge Av_k and expand the definition linearly to all tensors. More generally, we can define \bigwedge^p A^k \in \operatorname\bigl(\bigwedge^p V\bigr), (p \geq k) on simple tensors by \begin &\left(^p A^k \right)(v_1 \wedge \cdots \wedge v_p) \\ 0mu&\qquad= \sum_ v_1 \wedge \cdots \wedge Av_ \wedge \cdots \wedge Av_ \wedge \cdots \wedge v_p \end i.e. choose ''k'' components on which ''A'' would act, then sum up all results obtained from different choices. If p < k, define \bigwedge^p A^k = 0. Since \bigwedge^n V is 1-dimensional with basis e_1 \wedge \cdots \wedge e_n, we can identify \bigwedge^n A^k with the unique number \kappa \in K satisfying ^n A^k (e_1 \wedge \cdots \wedge e_n) = \kappa (e_1 \wedge \cdots \wedge e_n) . * For \varphi \in \operatorname\bigl(^p V\bigr), define the exterior transpose \varphi^\mathrm \in \operatorname\bigl(\bigwedge^ V\bigr) to be the unique operator satisfying (\varphi^\mathrm\omega_) \wedge \omega_p = \omega_ \wedge (\varphi \omega_p) for any \omega_p \in ^p V and \omega_ \in ^V. * For A \in \operatorname(V), define \det A = \bigwedge^n A^n, \operatorname(A) = \bigwedge^n A^1, \operatorname A = \bigl(\bigwedge^ A^\bigr)^\mathrm. These are equivalent to the previous definitions.


Basic properties

All results obtained from other definitions of the determinant, trace and adjoint can be obtained from this definition (since these definitions are equivalent). Here are some basic properties related to these new definitions: * (\cdot)^\mathrm is K -linear. * (AB)^\mathrm = B^\mathrm A^\mathrm. * We have a canonical isomorphism \begin \psi:\operatorname\bigl(^k V\bigr) \cong \operatorname\bigl(^ V\bigr) \\ A \mapsto A^\mathrm \end However, there is no canonical isomorphism between \bigwedge^k V and \bigwedge^ V. * \operatorname \bigl(\bigwedge^k A \bigr) = \bigwedge^n A^k. The entries of the transposed matrix of \bigwedge^k A are k \times k -minors of A. * For all k \leq n-1, p \leq k, A \in \operatorname(V), \sum_^p \bigl(^ A^ \bigr)^\mathrm \bigl(^k A^q \bigr) = \bigl(^n A^p \bigr) \operatorname \in \operatorname(V). In particular, \bigl(^ A^ \bigr)^\mathrm A + \bigl(^ A^p \bigr)^\mathrm = \bigl(^n A^p \bigr) \operatorname and hence (\operatorname A)A = \bigl(^ A^ \bigr)^\mathrm A = \bigl(^n A^n \bigr) \operatorname =(\det A)\operatorname. * \bigl(^ A^p \bigr)^\mathrm = \sum_^p \bigl(^n A^ \bigr)(-A)^q = \sum_^p \operatorname \bigl(^ A \bigr)(-A)^q. In particular, \operatorname A = \sum_^ \bigl(^n A^ \bigr)(-A)^q. * \operatorname \bigl(^k \operatorname A \bigr) = ^n (\operatorname A)^k = (\det A)^ \bigl(^n A^ \bigr) = (\det A)^ \operatorname \bigl(^ A \bigr). * \operatorname\! \Bigl( \bigl(^ A^k \bigr)^\mathrm \Bigr) = (n-k) ^n A^p = (n-k) \operatorname\left(^p A \right). * The characteristic polynomial \operatorname_A(t) of A \in \operatorname(V) can be given by \operatorname_A(t) = \sum_^n \operatorname \bigl(^k A \bigr)(-t)^ = \sum_^n \left(^n A^k \right)(-t)^ . Similarly, \operatorname_(t) = \sum_^n \left(^n(\operatorname A)^k \right)(-t)^ = \sum_^n (\det A)^ \left(^n A^ \right)(-t)^


Leverrier's algorithm

\bigwedge^n A^k are the coefficients of the (-t)^ terms in the characteristic polynomial. They also appear in the expressions of \bigl(\bigwedge^ A^p\bigr)^\mathrm and \bigwedge^n (\operatorname A)^k. Leverrier's Algorithm is an economical way of computing \bigwedge^n A^k and \bigwedge^ A^k \colon :Set \bigwedge^ A^0 = 1; :For k = n-1, n-2, \ldots, 1, 0, :: ^n A^ = \frac \operatorname(A \circ ^ A^); :: ^ A^ = ^n A^ \cdot \operatorname - A \circ ^ A^.


Physics

In physics, many quantities are naturally represented by alternating operators. For 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. Its six degrees of freedom are identified with the electric and magnetic fields.


Linear geometry

The decomposable ''k''-vectors have geometric interpretations: the bivector represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
with sides ''u'' and ''v''. Analogously, the 3-vector represents the spanned 3-space weighted by the volume of the oriented
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...
with edges ''u'', ''v'', and ''w''.


Projective geometry

Decomposable ''k''-vectors in \bigwedge\nolimits^k V correspond to weighted ''k''-dimensional
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
s of ''V''. In particular, the Grassmannian of ''k''-dimensional subspaces of ''V'', denoted Gr''k''(''V''), can be naturally identified with an
algebraic subvariety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
of the
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
P\bigl(\bigwedge\nolimits^k V\bigr). This is called the Plücker embedding.


Differential geometry

The exterior algebra has notable applications in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, where it is used to define
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s. Differential forms are mathematical objects that evaluate the length of vectors, areas of parallelograms, and volumes of higher-dimensional bodies, so they can be integrated over curves, surfaces and higher dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s in a way that generalizes the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
s and
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
s from calculus. A
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
at a point of a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is an alternating multilinear form on the tangent space at the point. Equivalently, a differential form of degree ''k'' is a linear functional on the ''k''-th exterior power of the tangent space. As a consequence, the exterior product of multilinear forms defines a natural exterior product for differential forms. Differential forms play a major role in diverse areas of differential geometry. An alternate approach defines differential forms in terms of germs of functions. In particular, the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
gives the exterior algebra of differential forms on a manifold the structure of a
differential graded algebra In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure. __TOC__ Definition A differential graded a ...
. The exterior derivative commutes with
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
along smooth mappings between manifolds, and it is therefore a
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
. The exterior algebra of differential forms, equipped with the exterior derivative, is a cochain complex whose cohomology is called the
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
of the underlying manifold and plays a vital role in the
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
of differentiable manifolds.


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 representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _ ...
s of the general linear group; see fundamental representation.


Superspace

The exterior algebra over the complex numbers is the archetypal example of a
superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. T ...
, which plays a fundamental role in physical theories pertaining to
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s and
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
. A single element of the exterior algebra is called a supernumber or Grassmann number. The exterior algebra itself is then just a one-dimensional superspace: it is just the set of all of the points in the exterior algebra. The topology on this space is essentially the weak topology, the open sets being the
cylinder set In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. General definition Given a collection S of sets, consider the Cartesian product X = \prod_ ...
s. An -dimensional superspace is just the -fold product of exterior algebras.


Lie algebra homology

Let ''L'' be a Lie algebra over a field ''K'', then it is possible to define the structure of a chain complex on the exterior algebra of ''L''. This is a ''K''-linear mapping : \partial : ^L \to ^p L defined on decomposable elements by : \partial (x_1 \wedge \cdots \wedge x_) = \frac\sum_(-1)^ _j,x_\ell\wedge x_1 \wedge \cdots \wedge \hat_j \wedge \cdots \wedge \hat_\ell \wedge \cdots \wedge x_. The Jacobi identity holds if and only if , and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra ''L'' to be a Lie algebra. Moreover, in that case \bigwedge L is a chain complex with boundary operator ∂. The homology associated to this complex is the Lie algebra homology.


Homological algebra

The exterior algebra is the main ingredient in the construction of the Koszul complex, a fundamental object in
homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topolo ...
.


History

The exterior algebra was first introduced by
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
in 1844 under the blanket term of ''Ausdehnungslehre'', or ''Theory of Extension''. 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. Saint-Venant also published similar ideas of exterior calculus for which he claimed priority over Grassmann. 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 Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
, except focused exclusively on the task of formal reasoning in geometrical terms. 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. The import of this new theory of vectors and
multivector In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors ...
s was lost to mid 19th century mathematicians,. 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 Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
,
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
, and Gaston Darboux) who applied Grassmann's ideas to the calculus of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s. A short while later,
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applica ...
, borrowing from the ideas of Peano and Grassmann, introduced his
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study ...
. This then paved the way for the 20th century developments of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
by placing the axiomatic notion of an algebraic system on a firm logical footing.


See also

* Alternating algebra *
Exterior calculus identities This article summarizes several identities in exterior calculus. Notation The following summarizes short definitions and notations that are used in this article. Manifold M, N are n-dimensional smooth manifolds, where n\in \mathbb . That ...
*
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
, a generalization of exterior algebra using a nonzero
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
*
Geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...
* Koszul complex * Multilinear algebra * Symmetric algebra, the symmetric analog * Tensor algebra * Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic form


Notes


References


Mathematical references

* *: 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. * *: 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 §III.7 and §III.11. * *: This book contains applications of exterior algebras to problems in
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
. Rank and related concepts are developed in the early chapters. * *: 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.


Historical references

* * * * (The Linear Extension Theory – A new Branch of Mathematics
alternative reference
* * ; . *


Other references and further reading

* *: An introduction to the exterior algebra, and
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...
, with a focus on applications. Also includes a history section and bibliography. * *: Includes applications of the exterior algebra to differential forms, specifically focused on integration and Stokes's theorem. The notation \bigwedge\nolimits^k V in this text is used to mean the space of alternating ''k''-forms on ''V''; i.e., for Spivak \bigwedge\nolimits^k V is what this article would call \bigwedge\nolimits^k V^*. Spivak discusses this in Addendum 4. * *: Includes an elementary treatment of the axiomatization of determinants as signed areas, volumes, and higher-dimensional volumes. * * *: This textbook in
multivariate calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather th ...
introduces the exterior algebra of differential forms adroitly into the calculus sequence for colleges. * *: An introduction to the coordinate-free approach in basic finite-dimensional linear algebra, using exterior products. * *: Chapter 10: The Exterior Product and Exterior Algebras
"The Grassmann method in projective geometry"
A compilation of English translations of three notes by Cesare Burali-Forti on the application of exterior algebra to projective geometry
C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"
An English translation of an early book on the geometric applications of exterior algebras
"Mechanics, according to the principles of the theory of extension"
An English translation of one Grassmann's papers on the applications of exterior algebra {{tensors Algebras Multilinear algebra Differential forms