Difference between revisions of "Basis"
Jump to navigation
Jump to search
(deleted confusing and inappropriate comparison to vector space bases) |
(definition in linear algebra) |
||
| Line 1: | Line 1: | ||
''Basis is a [[mathematics]] term.'' | ''Basis is a [[mathematics]] term.'' | ||
| + | |||
| + | == Linear algebra == | ||
| + | |||
| + | In [[linear algebra]], a basis is a set of [[linearly independent]] vectors that [[spans]] a vector space. 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 | A '''basis''' ''B'' for a [[topology]] ''T'' on a set ''X'' is a collection of subsets of ''X'' (called '''basis elements''') such that | ||
Revision as of 02:17, April 24, 2010
Basis is a mathematics term.
Linear algebra
In linear algebra, a basis is a set of linearly independent vectors that spans a vector space. 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 ∩ 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.