In multilinear algebra, a multivector, sometimes called Clifford number or multor, 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 ∧ ⋯ ∧ v k , {\displaystyle v_{1}\wedge \cdots \wedge v_{k},}
where v 1 , … , v k {\displaystyle 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 with potentially differing values of k). In differential geometry, a k-vector is usually a vector in the exterior algebra of the tangent vector space of a smooth manifold; that is, it is an antisymmetric tensor obtained by taking linear combinations of the exterior product of k tangent vectors, for some integer k ≥ 0. A differential 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.
Exterior product
The exterior product (also called the wedge product) used to construct multivectors is multilinear (linear in each input), associative and alternating. This means for vectors u, v and w in a vector space V and for scalars α, β, the exterior product has the properties:
Linear in an input: u ∧ ( α v + β w ) = α u ∧ v + β u ∧ w ; {\displaystyle \mathbf {u} \wedge (\alpha \mathbf {v} +\beta \mathbf {w} )=\alpha \mathbf {u} \wedge \mathbf {v} +\beta \mathbf {u} \wedge \mathbf {w} ;}
Associative: ( u ∧ v ) ∧ w = u ∧ ( v ∧ w ) ; {\displaystyle (\mathbf {u} \wedge \mathbf {v} )\wedge \mathbf {w} =\mathbf {u} \wedge (\mathbf {v} \wedge \mathbf {w} );}
Alternating: u ∧ u = 0. {\displaystyle \mathbf {u} \wedge \mathbf {u} =0.}
The exterior product of k vectors or a sum of such products (for a single k) is called a grade k multivector, or a k-vector. The maximum grade of a multivector is the dimension of the vector space V. Linearity in either input together with the alternating property implies linearity in the other input. The multilinearity of the exterior product allows a multivector to be expressed as a linear combination of exterior products of basis vectors of V. The exterior product of k basis vectors of V is the standard way of constructing each basis element for the space of k-vectors, which has dimension (nk) in the exterior algebra of an n-dimensional vector space.
Area and volume The k-vector obtained from the exterior product of k separate vectors in an n-dimensional space has components that define the projected (k − 1)-volumes of the k-parallelotope spanned by the vectors. The square root of the sum of the squares of these components defines the volume of the k-parallelotope. The following examples show that a bivector in two dimensions measures the area of a parallelogram, and the magnitude of a bivector in three dimensions also measures the area of a parallelogram. Similarly, a three-vector in three dimensions measures the volume of a parallelepiped. It is easy to check that the magnitude of a three-vector in four dimensions measures the volume of the parallelepiped spanned by these vectors.
Multivectors in R2 Properties of multivectors can be seen by considering the two-dimensional vector space V = R2. Let the basis vectors be e1 and e2, so u and v are given by
u = u 1 e 1 + u 2 e 2 , v = v 1 e 1 + v 2 e 2 , {\displaystyle \mathbf {u} =u_{1}\mathbf {e} _{1}+u_{2}\mathbf {e} _{2},\quad \mathbf {v} =v_{1}\mathbf {e} _{1}+v_{2}\mathbf {e} _{2},}
and the multivector u ∧ v, also called a bivector, is computed to be
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