kVektor

kVektor, or k-vector, is a concept in linear algebra describing an element of the k-th exterior power of a finite-dimensional vector space. Let V be a vector space over a field F (typically the real or complex numbers). The exterior power ∧^k V comprises all k-vectors. For k = 0, a 0-vector is a scalar; for k = 1, it is a ordinary vector; for higher k, k-vectors generalize antisymmetric multilinear quantities such as areas and volumes.

If the dimension of V is n, then ∧^k V has dimension C(n, k) (the binomial coefficient). A basis is given by wedge products e_i1 ∧ … ∧ e_ik where {e1, …, en} is a basis of V and i1 < … < ik. A k-vector is decomposable (simple) when it can be written as a single wedge product v1 ∧ … ∧ vk; in general, a k-vector may be a sum of decomposable terms.

The wedge product is bilinear and antisymmetric: swapping two factors changes the sign, and in particular v ∧ w = - w ∧ v. This antisymmetry underpins many properties of k-vectors and their applications.

Coordinate representation and applications: differential forms are sections of ∧^k T* M on a manifold M, and wedge products enable compact expressions of orientation and volume. Determinants can be interpreted through wedge products of basis vectors, and k-vectors appear in physics as antisymmetric tensor fields and in the description of oriented volumes and integration on manifolds.

History: the exterior algebra was introduced by Hermann Grassmann in the 19th century and was later developed by Elie Cartan and others. Today, k-vectors are a fundamental tool in differential geometry, multilinear algebra, and mathematical physics.