Span (mathematics): Difference between revisions
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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]].[[Category:Suggestion Bot Tag]] |
Latest revision as of 17:00, 20 October 2024
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.