Algebre: Chapitre 10.Algebre homologique by N. Bourbaki

By N. Bourbaki

Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.

Ce dixième chapitre du Livre d Algèbre, deuxième Livre du traité, pose les bases du calcul homologique.

Ce quantity est a été publié en 1980.

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Then there is a basis B' of V with B � B' � A U B . Again, this theorem i s true generally even i n the infinite case, but requires more sophisticated set theory to prove. 21 also provides a method for calculating a suitable basis. For example, suppose V is the real vector space JR4 and a = ( 1 , 1, 0 , 0)T, b = ( 1, 1 , 1 , 1)T. Then the set B = {a, b} is linearly independent so can be extended to a basis. To find such a basis, start with the usual basis vectors e 1 = ( 1 , 0, 0, 0) r , e 2 = (0, 1, 0, 0) T , e3 = (0, 0, 1, 0) r, e4 = (0, 0 , 0 , 1) T of IR4 .

42 Let F = IF2 F8 = = {0, 1 } , the field of order 2. Then { (x1 , . . , xs ) T : Xi E F} is a vector space of dimension 8 over F, with a basis { (1 , o, o, o, o, o, o, of, (o, 1 , o, o, o, o, o, of, . . , (O, o, o, o, o, o, o, 1)r}. These vectors are very important i n computer science, where they are called 'bytes'. The number of such vectors is 28 , as there are two possibilities for x1 , two possibilities for x 2 , and so on. More generally, if F is a field of order then contains exactly vectors.

For the case of a vector space over C, the argument is the same, but use D scalars from C instead. as as It follows that any two real vector spaces V, W of dimension n are isomorphic, are any two complex vector spaces V, W of dimension n . 6 Vector spaces over other fields The observant reader might have noticed that the two kinds of vector spaces we have been considering-over the reals and over the complexes-have much in common and he or she may wonder whether the notion of a vector space makes sense over any other number system other than � or C.

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