Vectorv

Vectorv is a term used in some mathematical discussions to denote an augmented vector, defined as an ordered pair (v, a) where v is a conventional vector in R^n and a is a scalar in R. The collection of all vectorvs is the Cartesian product R^n × R, which is itself a real vector space of dimension n+1 and is isomorphic to R^{n+1}.

Operations on vectorvs are performed componentwise: (v1,a1) + (v2,a2) = (v1+v2, a1+a2) and α·(v,a) = (α v, α a) for scalar α. A common inner product on vectorv spaces is ⟨(v1,a1),(v2,a2)⟩ = ⟨v1,v2⟩ + β a1 a2, where ⟨v1,v2⟩ is the standard inner product on R^n and β > 0 is a fixed weight. The associated norm is ||(v,a)|| = sqrt(||v||^2 + β a^2).

In practice, vectorv can be used to encode both position and a scalar attribute, such as intensity, time, or weight, for each element in a geometry or dataset. This makes vectorv suitable as a feature representation in computer graphics, computer vision, and certain machine learning contexts where a per-element scalar attribute is coupled with a spatial vector.

Variants include choosing different base fields (e.g., C^n or finite fields) or different coupling in the inner product. When n is 2 or 3, vectorv representations align with augmented 2D or 3D coordinates (for example, (x,y,t) or (x,y,z,t)).

See also: augmented vector, product space, feature vector, multivariate data.