Multivector

Element of an exterior algebra

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 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 with potentially differing values of k). In differential geometry, a k vector is a vector in the exterior algebra of the tangent vector space; 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 1=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.

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.

In this section, we consider multivectors on a projective space Pn, which provide a convenient set of coordinates for lines, planes and hyperplanes that have properties similar to the homogeneous coordinates of points, called Grassmann coordinates. Points in a real projective space Pn are defined to be lines through the origin of the vector space Rn+1. For example, the projective plane P2 is the set of lines through the origin of R3. Thus, multivectors defined on Rn+1 can be viewed as multivectors on Pn. A convenient way to view a multivector on Pn is to examine it in an affine component of Pn, which is the intersection of the lines through the origin of Rn+1 with a selected hyperplane, such as 1=H: xn+1 = 1. Lines through the origin of R3 intersect the plane 1=E: z = 1 to define an affine version of the projective plane that only lacks the points for which 1=z = 0, called the points at infinity.

Bivectors play many important roles in physics, for example, in the classification of electromagnetic fields.