Multivector

In <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a>, a multivector, sometimes called Clifford number or multor, is an element of the <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> Λ(V) of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V. This algebra is <a href="/facts/Graded_algebra/AIUjkSaP">graded</a>, <a href="/facts/Associative_algebra/DRfoF6KV">associative</a> and <a href="/facts/Alternating_algebra/kDoTSQXn">alternating</a>, and consists of <a href="/facts/Linear_combination/CVjL6kuN">linear combinations</a> of simple k-vectors (also known as decomposable k-vectors or <a href="/facts/Blade_(geometry)/0WY05nWt">k-blades</a>) 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 <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, a k-vector is usually a vector in the exterior algebra of the <a href="/facts/Tangent_space/crvpbSeG">tangent vector space</a> of a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a>; that is, it is an antisymmetric <a href="/facts/Tensor/pNjFo5tV">tensor</a> obtained by taking linear combinations of the <a href="/facts/Exterior_product/rdN4TvpR">exterior product</a> of k <a href="/facts/Tangent_vector/UEq5yffm">tangent vectors</a>, for some integer k ≥ 0. A <a href="/facts/Differential_form/T9gyHtMv">differential k-form</a> is a k-vector in the exterior algebra of the <a href="/facts/Dual_space/4Uw4Knz1">dual</a> 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 <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a>, <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vectors</a>, <a href="/facts/Bivector/PBRcVE8q">bivectors</a> and trivectors; they are respectively dual to <a href="/facts/Differential_form/T9gyHtMv">0-forms, 1-forms, 2-forms and 3-forms</a>.

Multivector open-in-new

Multivector