-vector, over a vector space
An element of the -th exterior degree of the space over a field (see Exterior algebra). A -vector can be understood as a -times skew-symmetrized contravariant tensor on . Any linearly independent system of vectors from defines a non-zero -vector ; such a poly-vector is called factorable, decomposable, pure, or prime (often simply a poly-vector). Here two linearly independent systems and generate the same subspace in if and only if , where . For any non-zero poly-vector , its annihilator is a subspace of dimension , and the poly-vector is pure if and only if . The pure -vectors of an -dimensional space form an algebraic variety in ; the corresponding projective algebraic variety is a Grassmann manifold. Any non-zero -vector or -vector in an -dimensional space is pure, but a bivector is pure if and only if .
If is a basis of and , then the coordinates of the poly-vector in the basis of the space are the minors , , of the matrix . In particular, for ,
If one specifies a non-zero -vector , a duality between -vectors and -vectors is obtained, i.e. a natural isomorphism
such that for all and .
Let and let an inner product be defined in , then in an inner product is induced with the following property: For any orthonormal basis in the basis in is also orthonormal. The scalar square
of a pure poly-vector coincides with the square of the volume of the parallelopipedon in constructed on the vectors . If one specifies an orientation in the -dimensional Euclidean space (which is equivalent to choosing an -vector for which ), then the above duality leads to a natural isomorphism . In particular, the -vector corresponds to a vector , called the vector product of the vectors .
|||N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)|
|||A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian)|
|||M.M. Postnikov, "Linear algebra and differential geometry" , Moscow (1979) (In Russian)|
Poly-vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poly-vector&oldid=17085