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In mathematics, the linear span (also called the linear hull [1] or just span) of a set of elements of a vector space is the smallest linear subspace of that contains . It is the set of all finite linear combinations of the elements of S , [ 2 ] and the intersection of all linear subspaces that contain S . {\displaystyle S.}
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace [1] [note 1] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces .
The closure property also implies that every intersection of linear subspaces is a linear subspace. [11] Linear span Given a subset G of a vector space V, the linear span or simply the span of G is the smallest linear subspace of V that contains G, in the sense that it is the intersection of all linear subspaces that contain G.
In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation .
In linear algebra, the closure of a non-empty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset. It is a vector space by the preceding general result, and it can be proved easily that is the set of linear combinations of elements of the subset.
tends to 0 when n → ∞, where F n is the linear span of the basis vectors e m for m ≥ n. The unit vector basis for ℓ p, 1 < p < ∞, or for c 0, is shrinking. It is not shrinking in ℓ 1: if f is the bounded linear functional on ℓ 1 given by
A linear dependency or linear relation among vectors v 1, ..., v n is a tuple (a 1, ..., a n) with n scalar components such that + + =. If such a linear dependence exists with at least a nonzero component, then the n vectors are linearly
SL – special linear group. SO – special orthogonal group. SOC – second order condition. Soln – solution. Sp – symplectic group. Sp – trace of a matrix, from the German "spur" used for the trace. sp, span – linear span of a set of vectors. (Also written with angle brackets.) Spec – spectrum of a ring. Spin – spin group.