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This is an illustration of the closest vector problem (basis vectors in blue, external vector in green, closest vector in red). In CVP, a basis of a vector space V and a metric M (often L 2) are given for a lattice L, as well as a vector v in V but not necessarily in L. It is desired to find the vector in L closest to v (as measured by M).
With the conditions and definitions above, and some manipulation, it can be shown that the term () =, which then justifies the previous solution of the normal to a plane problem. Since the vector term of the vector bivector product the name dot product is zero when the vector is perpendicular to the plane (bivector), and this vector, bivector ...
A vector pointing from point A to point B. In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector [1] or spatial vector [2]) is a geometric object that has magnitude (or length) and direction.
The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Bregman–Minc inequality (discrete mathematics) Cauchy-Binet formula (linear algebra) Cayley–Hamilton theorem (Linear algebra) Dimension theorem for vector spaces (vector spaces, linear algebra) Euler's rotation theorem ; Exchange theorem (linear algebra) Gamas's Theorem (multilinear algebra) Gershgorin circle theorem (matrix theory)
Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2 w . In mathematics and physics , a vector space (also called a linear space ) is a set whose elements, often called vectors , can be added together and multiplied ...
In mathematics, vector algebra may mean: The operations of vector addition and scalar multiplication of a vector space; The algebraic operations in vector calculus (vector analysis) – including the dot and cross products of 3-dimensional Euclidean space; Algebra over a field – a vector space equipped with a bilinear product