Unformatted text preview: ˛.a 1 ;:::;a s / D .˛a 1 ;:::;˛a s / 7! ˛a 1 C ±±± C ˛a s D ˛.a 1 C ±±± C a 1 C ±±± C a s /: n If the conditions of the proposition are satisﬁed, we say that V is the internal direct sum of the subspaces A i , and we write V D A 1 ˚±±±˚ A s . In particular, if M and N are subspaces of V such that M C N D V and M \ N D f g , then V D M ˚ N . Let N be a subspace of a vector space V . A subspace M of V is said to be a complement of M if V D M ˚ N . Subspaces of ﬁnite–dimensional vector spaces always have a complement, as we shall now explain. Proposition 3.3.34. Let T W V ! W be a surjective linear map of a ﬁnite–dimensional vector space V onto a vector space W . Then T admits a right inverse; that is, there exists a linear map S W W ! V such that T ı S D id W ....
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 Fall '08
 EVERAGE
 Algebra, Vector Space, 2 K

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