Difference between revisions of "Basis"
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''Basis is a [[mathematics]] term.'' | ''Basis is a [[mathematics]] term.'' | ||
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| + | == Linear algebra == | ||
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| + | In [[linear algebra]], a basis is a set of [[linear independence|linearly independent]] vectors that [[spans]] a vector space ''V''. Any vector in the vector space can then be written as a linear combination of the basis. | ||
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| + | This usage of the term is similar to its common usage: a basis is the foundation for what is needed. | ||
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| + | == Topology == | ||
A '''basis''' ''B'' for a [[topology]] ''T'' on a set ''X'' is a collection of subsets of ''X'' (called '''basis elements''') such that | A '''basis''' ''B'' for a [[topology]] ''T'' on a set ''X'' is a collection of subsets of ''X'' (called '''basis elements''') such that | ||
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If ''B'' satisfy the above 2 conditions, then the '''topology ''T'' generated by ''B''''' is the collection of subsets ''U'' of ''X'' such that for each ''x'' in ''U'', there is a basis element ''V'' in ''B'' such that ''x'' is in ''V'' and ''V'' is a subset of ''U''. | If ''B'' satisfy the above 2 conditions, then the '''topology ''T'' generated by ''B''''' is the collection of subsets ''U'' of ''X'' such that for each ''x'' in ''U'', there is a basis element ''V'' in ''B'' such that ''x'' is in ''V'' and ''V'' is a subset of ''U''. | ||
| − | + | [[Category:Topology]][[Category:Linear Algebra]] | |
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Latest revision as of 15:57, September 11, 2017
Basis is a mathematics term.
Linear algebra
In linear algebra, a basis is a set of linearly independent vectors that spans a vector space V. Any vector in the vector space can then be written as a linear combination of the basis.
This usage of the term is similar to its common usage: a basis is the foundation for what is needed.
Topology
A basis B for a topology T on a set X is a collection of subsets of X (called basis elements) such that
- each x in X, is in at least one basis element.
- if x is in the intersection of 2 basis elements B1 and B2, then it is in some basis element B3, where B3 is a subset of B1 â© B2.
If B satisfy the above 2 conditions, then the topology T generated by B is the collection of subsets U of X such that for each x in U, there is a basis element V in B such that x is in V and V is a subset of U.