Span (mathematics): Difference between revisions
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In [[algebra]], the '''span''' of a set of elements of a [[module (algebra)|module]] or [[vector space]] is the set of all finite [[linear combination]]s of that set: it may equivalently be defined as the [[intersection]] of all [[submodule]]s or [[subspace]]s containing the given set. | In [[algebra]], the '''span''' of a set of elements of a [[module (algebra)|module]] or [[vector space]] is the set of all finite [[linear combination]]s of that set: it may equivalently be defined as the [[intersection]] of all [[submodule]]s or [[subspace]]s containing the given set. | ||
For ''S'' a [[subset]] of an ''R''-module ''M'' we have | For ''S'' a [[subset]] of an ''R''-module ''M'' we have | ||
:<math>\langle S \rangle = \left\lbrace \sum_{i=1}^n r_i s_i : r_i \in R,~ s_i \in S \right\rbrace = \bigcap_{S \subseteq N; N \le M | :<math>\langle S \rangle = \left\lbrace \sum_{i=1}^n r_i s_i : r_i \in R,~ s_i \in S \right\rbrace = \bigcap_{S \subseteq N; N \le M} N .\,</math> | ||
We say that ''S'' spans, or is a '''spanning set''' for <math>\langle S \rangle</math>. | |||
A [[Basis (linear algebra)|basis]] is a [[Linear independence|linearly independent]] spanning set. | |||
If ''S'' is itself a submodule then <math>S = \langle S \rangle</math>. | If ''S'' is itself a submodule then <math>S = \langle S \rangle</math>. | ||
The equivalence of the two definitions follows from the property of the submodules forming a [[closure system]] for which <math>\langle \cdot \rangle</math> is the corresponding [[closure operator]]. | The equivalence of the two definitions follows from the property of the submodules forming a [[closure system]] for which <math>\langle \cdot \rangle</math> is the corresponding [[closure operator]]. | ||
Latest revision as of 13:20, 7 February 2009
In algebra, the span of a set of elements of a module or vector space is the set of all finite linear combinations of that set: it may equivalently be defined as the intersection of all submodules or subspaces containing the given set.
For S a subset of an R-module M we have
We say that S spans, or is a spanning set for .
A basis is a linearly independent spanning set.
If S is itself a submodule then .
The equivalence of the two definitions follows from the property of the submodules forming a closure system for which is the corresponding closure operator.
