# Themes Navi

Sep 28, 2017 - 34 minute read

# Incontri olimpici algebra

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 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 bakecha incontri la 1 are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the incontri olimpici algebra generated by those 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 incontri olimpici algebra concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra incontri olimpici algebra objects that are not only k -blades, but sums of k -blades; such a sum luoghi incontri udine 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 vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials.