In multilinear algebra, a multivector, sometimes called Clifford number, 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 (also known as decomposable k-vectors 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).
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.
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,