College Algebra Exam Review 160

College Algebra Exam Review 160 - ˛.a 1;a s D.˛a 1;˛a s...

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170 3. PRODUCTS OF GROUPS and ˛.a 1 ;a 2 ;:::;a n / D .˛a 1 ;˛a 2 ;:::;˛a n /; for a i ;b i 2 V i and ˛ 2 K . The direct sum is denoted by V 1 ˚ V 2 ˚±±±˚ V n . How can we recognize that a vector space V is isomorphic to the direct sum of several subspaces A 1 ;A 2 ;:::;A n ? It is neccessary and sufﬁcient that V be be isomorphic to the direct product of the A i , regarded as abelian groups. Proposition 3.3.33. Let V be a vector space over a ﬁeld K with subspaces A 1 ;:::A s such that V D A 1 C ±±± C A s . Then the following conditions are equivalent: (a) .a 1 ;:::;a s / 7! a 1 C ±±± C a s is a group isomorphism of A 1 ² ±±± ² A s onto V . (b) .a 1 ;:::;a s / 7! a 1 C ±±± C a s is a linear isomorphism of A 1 ˚ ±±± ˚ A s onto V . (c) Each element x 2 V can be expressed as a sum x D a 1 C±±±C a s , with a i 2 A i for all i , in exactly one way. (d) If 0 D a 1 C ±±± C a s , with a i 2 A i for all i , then a i D 0 for all i . Proof. The equivalence of (a), (c), and (d) is by Proposition 3.5.1 . Clearly (b) implies (a). We have only to show that if .a/ holds, then the map .a 1 ;:::;a s / 7! a 1 C ±±± C a s respects multiplication by elements of K . This is immediate from the computation
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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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