# Multivector

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

$v_{1}\wedge \cdots \wedge v_{k},$ where $v_{1},\ldots ,v_{k}$ 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.

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.

## 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: $\mathbf {u} \wedge (\alpha \mathbf {v} +\beta \mathbf {w} )=\alpha \mathbf {u} \wedge \mathbf {v} +\beta \mathbf {u} \wedge \mathbf {w} ;$ • Associative: