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 infinite (in the latter case, it is a cardinal number), and defines the dimension of the vector space. Formally, the dimension theorem for vector spaces states that Given a vector space V, any two bases have the same cardinality. As a basis is a generating set that is linearly independent, the theorem is a consequence of the following theorem, which is also useful:
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| - Teorema de la dimensió per espais vectorials (ca)
- Dimension theorem for vector spaces (en)
- Théorème de la dimension pour les espaces vectoriels (fr)
- Teorema della dimensione per spazi vettoriali (it)
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| - En mathématiques, le théorème de la dimension pour les espaces vectoriels énonce que deux bases quelconques d'un même espace vectoriel ont même cardinalité. Joint au théorème de la base incomplète qui assure l'existence de bases, il permet de définir la dimension d'un espace vectoriel comme le cardinal (fini ou infini) commun à toutes ses bases. (fr)
- In matematica, il teorema della dimensione per spazi vettoriali afferma che basi diverse di uno stesso spazio vettoriale hanno la stessa cardinalità, ovvero sono costituite dallo stesso numero di elementi. La cardinalità della base è inoltre pari alla dimensione dello spazio. In altri termini, sia uno spazio vettoriale su un campo . Siano e due basi di la cui dimensione sia rispettivamente e . Allora . (it)
- En matemàtiques, el teorema de la dimensió per espais vectorials afirma que totes les bases d'un espai vectorial tenen el mateix nombre d'elements. Aquest nombre d'elements pot ser finit, o bé un nombre cardinal infinit, que defineix la dimensió de l'espai vectorial. Formalment, el teorema de la dimensió per espais vectorials afirma que Donat un espai vectorial V, dos sistemes generadors linealment independents qualssevol (en altres paraules, dues bases qualssevol) tenen la mateixa cardinalitat. (ca)
- 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 infinite (in the latter case, it is a cardinal number), and defines the dimension of the vector space. Formally, the dimension theorem for vector spaces states that Given a vector space V, any two bases have the same cardinality. As a basis is a generating set that is linearly independent, the theorem is a consequence of the following theorem, which is also useful: (en)
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| - , (en)
- In a vector space , if is a generating set, and is a linearly independent set, then the cardinality of is not larger than the cardinality of . (en)
- Given a vector space , any two bases have the same cardinality. (en)
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| - En matemàtiques, el teorema de la dimensió per espais vectorials afirma que totes les bases d'un espai vectorial tenen el mateix nombre d'elements. Aquest nombre d'elements pot ser finit, o bé un nombre cardinal infinit, que defineix la dimensió de l'espai vectorial. Formalment, el teorema de la dimensió per espais vectorials afirma que Donat un espai vectorial V, dos sistemes generadors linealment independents qualssevol (en altres paraules, dues bases qualssevol) tenen la mateixa cardinalitat. Si V és un mòdul finitament generat, llavors té una base finita, i el resultat afirma que dues bases qualssevol tenen el mateix nombre d'elements. Mentre que la demostració de l'existència d'una base per qualsevol espai vectorial requereix el Lema de Zorn (equivalent a l'axioma de l'elecció), la unicitat de la cardinalitat de la base només necessita el lema de l'ultrafiltre, que és estrictament més feble; tot i això, la demostració que en donarem assumeix la , és a dir, que tots els nombres cardinals són comparables, una afirmació que és equivalent a l'axioma de l'elecció. Aquest teorema es pot generalitzar a R-mòduls amb . El teorema pel cas finitament generat no necessita l'axioma de l'elecció, sinó que es pot demostrar amb arguments bàsics de l'àlgebra lineal. (ca)
- 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 infinite (in the latter case, it is a cardinal number), and defines the dimension of the vector space. Formally, the dimension theorem for vector spaces states that Given a vector space V, any two bases have the same cardinality. As a basis is a generating set that is linearly independent, the theorem is a consequence of the following theorem, which is also useful: In a vector space V, if G is a generating set, and I is a linearly independent set, then the cardinality of I is not larger than the cardinality of G. In particular if V is finitely generated, then all its bases are finite and 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. In the finitely generated case the proof uses only elementary arguments of algebra, and does not require the axiom of choice nor its weaker variants. (en)
- En mathématiques, le théorème de la dimension pour les espaces vectoriels énonce que deux bases quelconques d'un même espace vectoriel ont même cardinalité. Joint au théorème de la base incomplète qui assure l'existence de bases, il permet de définir la dimension d'un espace vectoriel comme le cardinal (fini ou infini) commun à toutes ses bases. (fr)
- In matematica, il teorema della dimensione per spazi vettoriali afferma che basi diverse di uno stesso spazio vettoriale hanno la stessa cardinalità, ovvero sono costituite dallo stesso numero di elementi. La cardinalità della base è inoltre pari alla dimensione dello spazio. In altri termini, sia uno spazio vettoriale su un campo . Siano e due basi di la cui dimensione sia rispettivamente e . Allora . (it)
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