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In mathematics , the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas , volumes , and their higher-dimensional analogues. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation —a choice of clockwise or counterclockwise. 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 volume 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 , or Grassmann algebra after Hermann Grassmann ,  is the algebraic system whose product is the exterior product. 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 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. 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 polynomials. This means that the exterior algebra is a graded algebra.
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 over a commutative ring , and for other structures of interest in abstract algebra. 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 forms are mathematical objects that evaluate the length of infinitesimal vectors, areas of infinitesimal parallelograms, and volumes of higher-dimensional bodies , so they can be integrated over curves, surfaces and higher dimensional manifolds in a way that generalizes the line integrals and surface integrals from calculus. 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 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.
The Cartesian plane R 2 is a vector space equipped with a basis consisting of a pair of unit vectors.
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:. 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.
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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 of which the pair of vectors v and w form two adjacent sides, then A must satisfy the following properties:.
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 "standard" chosen parallelogram in a parallel plane here, the one with sides e 1 and e 2.
In other words, the exterior product provides a basis-independent formulation of area. For vectors in R 3 , the exterior algebra is closely related to the cross product and triple product.
The coefficients above are the same as those in the usual definition of the cross product of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a 2-vector. The scalar coefficient is the triple product of the three vectors. The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations.
It can also be interpreted as the vector consisting of the minors of the matrix with columns u and v. The exterior product in three dimensions allows for similar interpretations: it, too, can be identified with oriented areas or volumes spanned by the two or three vectors involved. In fact, in the presence of a positively oriented orthonormal basis , the exterior product generalizes these geometric notions to higher dimensions. That is, if.
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.
For example, in R 4 , the following 2-vector is not decomposable:. The reason is the following: given any exterior product of the form. 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 k -vectors can be computed as the minors of the matrix that describes the vectors v j in terms of the basis e i. Any element of the exterior algebra can be written as a sum of k -vectors. Hence, as a vector space the exterior algebra is a direct sum.
This is similar to the notion of tensor rank. As a consequence, the direct sum decomposition of the preceding section. In addition to studying the graded structure on the exterior algebra, Bourbaki 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.
Let V be a vector space over the field K. In other words, the exterior algebra has the following universal property : .
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 associative algebra that contains V , the tensor algebra T V , and then enforce the alternating property by taking a suitable quotient. This approach is often used in differential geometry and is described in the next section.
It will satisfy the analogous universal property. 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 Bourbaki Exterior algebras of vector bundles 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. Given two vector spaces V and X , an alternating operator from V k to X is a multilinear map.
In this case an alternating multilinear function. 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. In particular, the dimension of the space of alternating maps from V k to K is n choose k.
Under this identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones.
As in the case of tensor products of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables.
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It is defined as follows: . This definition of the exterior product is well-defined even if the field K has finite characteristic , if one considers an equivalent version of the above that does not use factorials or any constants:. The exterior algebra as well as the symmetric algebra inherits a bialgebra structure, and, indeed, a Hopf algebra structure, from the tensor algebra. See the article on tensor algebras for a detailed treatment of the topic.
The symbol 1 stands for the unit element of the field K. 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. This distinction is developed in greater detail in the article on tensor algebras.
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:. The coproduct and counit, along with the exterior product, define the structure of a bialgebra on the exterior algebra. Suppose that V is finite-dimensional. These three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case. Suppose that V has finite dimension n.
Applications of Grassmann's Extensive Algebra
Then the interior product induces a canonical isomorphism of vector spaces. 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. 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 of V.
In this case,. The pairing between these two spaces also takes the form of an inner product. On decomposable k -vectors,. It is not hard to show that for vectors v 1 ,v 2 , With respect to the inner product, exterior multiplication and the interior product are mutually adjoint. This property completely characterizes the inner product on the exterior algebra. Then, by the universal construction, there exists a unique homomorphism of graded algebras.
In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:. If K is a field of characteristic 0,  then the exterior algebra of a vector space V can be canonically identified with the vector subspace of T V consisting of antisymmetric tensors.
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Let T r V be the space of homogeneous tensors of degree r. This is spanned by decomposable tensors.