Vector Space
A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. An example of a vector space is that of Euclidean vectors, which may be used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessary have to be an arrow-like objects as they appear in the mentioned examples; one should think of these vectors as abstract mathematical objects which hold specific properties and in some cases, they can be visualized as arrows.
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Some articles on vector space:
... In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space V together with a subgroup G of GL(V) such that G has an open dense orbit in V ... Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and analysis, as well as representation theory ...
... Dimension of a vector space V over a field K is the cardinality of any Hamel basis of V ... For a linear subspace W of a vector space V we define codimension of W (with respect to V) ... degree equals the dimension of the extension as a vector space over the smaller field) ...
... In the first diagram the circled nodes represent 1-spaces and 2-spaces in a three-dimensional vector space ... In the second diagram the circled nodes represent 1-space and 2-spaces in a 3-dimensional vector space over the quaternions, which in turn represent certain 2-spaces and 4-spaces in a 6-dimensional complex ...
... A convex set can be seen as the combinations of the axioms for an affine space and a cone, which is reflected in the standard space for it, the n-simplex, being the intersection of the ... Such spaces are particularly used in linear programming ... In the language of universal algebra, a vector space is an algebra over the universal vector space K∞ of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an ...
... A Lie algebra structure on a vector space is a map which is skew-symmetric, and satisfies the Jacobi identity ... Dually, a Lie coalgebra structure on a vector space E is a linear map which is antisymmetric (this means that it satisfies, where is the canonical flip ... The dual space E* carries the structure of a bracket defined by α = dα(x∧y), for all α ∈ E and x,y ∈ E* ...
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