# Algebre: Chapitre 10.Algebre homologique by N. Bourbaki By N. Bourbaki

Ce dixiÃ¨me chapitre du Livre d AlgÃ¨bre, deuxiÃ¨me Livre du traitÃ©, pose les bases du calcul homologique.

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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.