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- Dimension_theorem_for_vector_spaces abstract "In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite, or given by an infinite cardinal number, and defines the dimension of the space.Formally, the dimension theorem for vector spaces states thatGiven a vector space V, any two linearly independent generating sets (in other words, any two bases) have the same cardinality.If V is finitely generated, then it has a finite basis, and the result says that any two bases have the same number of elements.While the proof of the existence of a basis for any vector space in the general case requires Zorn's lemma and is in fact equivalent to the axiom of choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma, which is strictly weaker (the proof given below, however, assumes trichotomy, i.e., that all cardinal numbers are comparable, a statement which is also equivalent to the axiom of choice). The theorem can be generalized to arbitrary R-modules for rings R having invariant basis number.The theorem for finitely generated case can be proved with elementary arguments of linear algebra, and requires no forms of the axiom of choice.".
- Dimension_theorem_for_vector_spaces wikiPageID "346992".
- Dimension_theorem_for_vector_spaces wikiPageRevisionID "548385613".
- Dimension_theorem_for_vector_spaces hasPhotoCollection Dimension_theorem_for_vector_spaces.
- Dimension_theorem_for_vector_spaces subject Category:Articles_containing_proofs.
- Dimension_theorem_for_vector_spaces subject Category:Theorems_in_abstract_algebra.
- Dimension_theorem_for_vector_spaces subject Category:Theorems_in_linear_algebra.
- Dimension_theorem_for_vector_spaces type Abstraction100002137.
- Dimension_theorem_for_vector_spaces type Communication100033020.
- Dimension_theorem_for_vector_spaces type Message106598915.
- Dimension_theorem_for_vector_spaces type Proposition106750804.
- Dimension_theorem_for_vector_spaces type Statement106722453.
- Dimension_theorem_for_vector_spaces type Theorem106752293.
- Dimension_theorem_for_vector_spaces type TheoremsInAbstractAlgebra.
- Dimension_theorem_for_vector_spaces type TheoremsInLinearAlgebra.
- Dimension_theorem_for_vector_spaces comment "In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements.".
- Dimension_theorem_for_vector_spaces label "Dimension theorem for vector spaces".
- Dimension_theorem_for_vector_spaces label "Teorema della dimensione per spazi vettoriali".
- Dimension_theorem_for_vector_spaces label "Théorème de la dimension pour les espaces vectoriels".
- Dimension_theorem_for_vector_spaces sameAs Théorème_de_la_dimension_pour_les_espaces_vectoriels.
- Dimension_theorem_for_vector_spaces sameAs Teorema_della_dimensione_per_spazi_vettoriali.
- Dimension_theorem_for_vector_spaces sameAs m.01ytsp.
- Dimension_theorem_for_vector_spaces sameAs Q3527205.
- Dimension_theorem_for_vector_spaces sameAs Q3527205.
- Dimension_theorem_for_vector_spaces sameAs Dimension_theorem_for_vector_spaces.
- Dimension_theorem_for_vector_spaces wasDerivedFrom Dimension_theorem_for_vector_spaces?oldid=548385613.
- Dimension_theorem_for_vector_spaces isPrimaryTopicOf Dimension_theorem_for_vector_spaces.