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Concept# Multivector

Summary

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 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 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 algebra
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:
Associative:
Alternating:
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.

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Bivector

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector can be thought of as being of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions.

Dyadics

In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context.

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 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).