Multivector

In multilinear algebra, a multivector, sometimes called Clifford number,[1] is an element of the exterior algebra Λ(V) of a vector space V. This algebra is graded, associative and alternating, and consists of linear combinations of simple k-vectors[2] (also known as decomposable k-vectors[3] or k-blades) of the form

where are in V.

A k-vector is such a linear combination that is homogeneous of degree k (all terms are k-blades for the same k). Depending on the authors, a "multivector" may be either a k-vector or any element of the exterior algebra (any linear combination of k-blades).[4]

In differential geometry, a k-vector is a k-vector in the exterior algebra of the tangent vector space; that is, it is an antisymmetric tensor obtained by taking linear combinations of the wedge product of k tangent vectors, for some integer k ≥ 0. A k-form is a k-vector in the exterior algebra of the dual of the tangent space, which is also the dual of the exterior algebra of the tangent space.

For k = 0, 1, 2 and 3, k-vectors are often called respectively scalars, vectors, bivectors and trivectors; they are respectively dual to 0-forms, 1-forms, 2-forms and 3-forms.[5][6]

Wedge product

The wedge product operation used to construct multivectors is linear, associative and alternating, which reflect the properties of the determinant. This means for vectors u, v and w in a vector space V and for scalars α, β, the wedge product has the properties,

  • Linear:
  • Associative: