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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 wikiPageLength "6692".
- Dimension_theorem_for_vector_spaces wikiPageOutDegree "30".
- Dimension_theorem_for_vector_spaces wikiPageRevisionID "548385613".
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Axiom_of_choice.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Basis_(linear_algebra).
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Boolean_prime_ideal_theorem.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Cardinal_number.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Cardinality.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Category:Articles_containing_proofs.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Category:Theorems_in_abstract_algebra.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Category:Theorems_in_linear_algebra.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Dimension_(vector_space).
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Finitely_generated_module.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Generator_(mathematics).
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Invariant_basis_number.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Kernel_(algebra).
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Linear_algebra.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Linear_independence.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Linear_map.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Linear_span.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Mathematics.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Module_(mathematics).
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Range_(mathematics).
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Rank_(linear_algebra).
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Rank–nullity_theorem.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Trichotomy_(mathematics).
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Tuple.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Vector_space.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Virtuous_circle_and_vicious_circle.
- Dimension_theorem_for_vector_spaces wikiPageWikiLink Zorns_lemma.
- Dimension_theorem_for_vector_spaces wikiPageWikiLinkText "Dimension theorem for vector spaces".
- Dimension_theorem_for_vector_spaces wikiPageWikiLinkText "dimension theorem for vector spaces".
- Dimension_theorem_for_vector_spaces wikiPageWikiLinkText "dimension theorem".
- 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 Proof.
- Dimension_theorem_for_vector_spaces type Theorem.
- 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 sameAs Q3527205.
- Dimension_theorem_for_vector_spaces sameAs Teorema_de_la_dimensió_per_espais_vectorials.
- 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 wasDerivedFrom Dimension_theorem_for_vector_spaces?oldid=548385613.
- Dimension_theorem_for_vector_spaces isPrimaryTopicOf Dimension_theorem_for_vector_spaces.