In mathematicsthe exterior product or wedge product of vectors is an algebraic construction used in geometry to study areasvolumesand their higher-dimensional analogues. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having **incontri olimpici algebra** same area, and with the same orientation —a choice of clockwise or counterclockwise. When regarded in this manner, *incontri olimpici algebra* 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 roma incontri pornostar vectors. The exterior algebraor Grassmann algebra after Hermann Grassmann[4] 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 kmeaning that they are sums of products of k *incontri olimpici algebra.*

In fact, in the presence of a positively oriented orthonormal basis , the exterior product generalizes these geometric notions to higher dimensions. Precedente Risultati delle Olimpiadi Internazionali di Matematica For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. Exterior algebras of vector bundles are frequently considered in geometry and topology. The interior product may also be described in index notation as follows. In applications to linear algebra , the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a matrix. 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. With respect to the inner product, exterior multiplication and the interior product are mutually adjoint. Here are some basic properties related to these new definitions:. 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.

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