# 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: