In mathematicsthe exterior product or wedge product of vectors is an algebraic construction used in geometry to study incontri olimpici algebravolumesand 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 incontri olimpici algebra 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 algebraor 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 incontri olimpici algebra.Navigation menu
Such an area is called the signed area of the parallelogram: The exterior algebra has notable applications in differential geometry , where it is used to define differential forms. 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. 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. 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 homology associated to this complex is the Lie algebra homology. The Road to Reality. The decomposable k -vectors have geometric interpretations: 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. There is a unique parallelogram having v and w as two of its sides. Winitzki, Lineaer Algebra via Exterior Products , https: These ideas can be extended not just to matrices but to linear transformations as well:
Esercizi di Algebra Incontri Olimpici - Montecatini Terme Esercizio 1. Sia p(x) un polinomio a coe cienti interi tale che p(1) = 7 e p(7) = 1. Incontri Olimpici Stage per Insegnanti su argomenti di matematica olimpica Dipartimento di Matematica "soulmind-online.com" - Viale Morgagni 67/A Firenze, Dicembre ALGEBRA Prof. Paolo Gronchi (Università di Firenze) Video Alessandra Caraceni (SNS, Pisa) Video. Gli Incontri Olimpici sono rivolti a docenti della scuola secondaria. Le quattro giornate sono dedicate ai quattro argomenti in cui possono essere suddivisi gli argomenti tipici delle competizioni matematiche: algebra, aritmetica (teoria dei numeri), combinatoria e geometria. Incontri Olimpici Stage per insegnanti su argomenti di matematica olimpica Aemilia Hotel - Bologna Lunedì 14/10 – Tema della giornata: ALGEBRA – Prof. Emanuele Callegari (Univ. di Roma “Tor Vergata”) – Prof. Devit Abriani (Univ. di Urbino).