P-vector

A multivector is the result of a product defined for elements in a vector space V. A vector space with a linear product operation between elements of the space is called an algebra; examples are matrix algebra and vector algebra. The algebra of multivectors is constructed using the wedge product ∧ and is related to the exterior algebra of differential forms.

The set of multivectors on a vector space V is graded by the number of basis vectors that form a basis multivector. A multivector that is the product of p basis vectors is called a grade p multivector, or a p-vector. The linear combination of basis p-vectors forms a vector space denoted as Λ^p(V). The maximum grade of a multivector is the dimension of the vector space V.

The product of a p-vector and a k-vector is a (k + p)-vector so the set of linear combinations of all multivectors on V is an associative algebra, which is closed with respect to the wedge product. This algebra, denoted by Λ(V), is called the exterior algebra of V.

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 (n
p) 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.

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