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 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 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). 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. 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, The product of p vectors is called a grade p multivector, or a p-vector. The maximum grade of a multivector is the dimension of the vector space V. The linearity of the wedge product allows a multivector to be defined as the linear combination of basis multivectors. There are (np) basis p-vectors in an n-dimensional vector space. The p-vector obtained from the wedge product of p separate vectors in an n-dimensional space has components that define the projected (p − 1)-volumes of the p-parallelotope spanned by the vectors. The square root of the sum of the squares of these components defines the volume of the p-parallelotope.