Zorn’s lemma and bases for vector spaces
Every linearly independent subset of can be extended to a basis for .
This has already been proved in this entry (http://planetmath.org/EveryVectorSpaceHasABasis). We reprove it here for completion.
Let be a linearly independent subset of . Let be the collection of all linearly independent supersets of . First, is non-empty since . In addition, if is a chain of linearly independent supersets of , then their union is again a linearly independent superset of (for a proof of this, see here (http://planetmath.org/PropertiesOfLinearIndependence)). So by Zorn’s Lemma, has a maximal element . Let . If , pick . If , where , then , so that . But , so , which implies . Consequently since is linearly independent. As a result, is a linearly independent superset of in , contradicting the maximality of in . ∎
Every spanning set of has a subset that is a basis for .
Let be a spanning set of . Let be the collection of all linearly independent subsets of . is non-empty as . Let be a chain of linearly independent subsets of . Then the union of these sets is again a linearly independent subset of . Therefore, by Zorn’s lemma, has a maximal element . In other words, is a linearly independent subset . Let . Suppose . Since spans , there is an element not in (for otherwise the span of must lie in , which would imply ). Then, using the same argument as in the previous proposition, is linearly independent, which contradicts the maximality of in . Therefore, spans and thus a basis for . ∎
Every vector space has a basis.
Either take to be the linearly independent subset of and apply proposition 1, or take to be the spanning subset of and apply proposition 2. ∎
Remark. The two propositions above can be combined into one: If are two subsets of a vector space such that is linearly independent and spans , then there exists a basis for , with . The proof again relies on Zorn’s Lemma and is left to the reader to try.
|Title||Zorn’s lemma and bases for vector spaces|
|Date of creation||2013-03-22 18:06:49|
|Last modified on||2013-03-22 18:06:49|
|Last modified by||CWoo (3771)|