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Any scalar product on a real vector space V is a pairing (set M = N = V, R = R in the above definitions). The determinant map (2 × 2 matrices over k ) → k can be seen as a pairing k 2 × k 2 → k {\displaystyle k^{2}\times k^{2}\to k} .
A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector.
In mathematics, a dual system, dual pair or a duality over a field is a triple (,,) consisting of two vector spaces, and , over and a non-degenerate bilinear map:. In mathematics , duality is the study of dual systems and is important in functional analysis .
A free object on X is a pair consisting of an object in C and an injection : (called the canonical injection), that satisfies the following universal property: For any object B in C and any map between sets g : X → U ( B ) {\displaystyle g:X\to U(B)} , there exists a unique morphism f : A → B {\displaystyle f:A\to B} in C such that g = U ...
A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement vector is a prototypical example of free vector. Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of metric.
Converting vector graphics to raster graphics. This terms also denotes a common method of rendering 3D models in real time. Ray casting Rendering by casting non-recursive rays from the camera into the scene. 2D ray casting is a 2.5D rendering method. Ray marching
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Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects).