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College Algebra Exam Review 341

College Algebra Exam Review 341 - sequence with j –entry...

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8.1. THE IDEA OF A MODULE 351 is actually a module homomorphism, because '.r.a 1 ; : : : ; a s // D '..ra 1 ; : : : ; ra n // D ra 1 C C ra s D r.a 1 C a s / D r'..a 1 ; : : : ; a s //: Therefore (a) implies (b). n Free modules Let R be a ring with identity element and let M be a (unital) R module. We define linear independence as for vector spaces: a subset S of M is linearly independent over R if whenever x 1 ; : : : ; x n are distinct elements of S and r 1 ; : : : ; r n are elements of R , if r 1 x 1 C r 2 x 2 C C r n x n D 0; then r i D 0 for all i . A basis for M is a linearly independent set S with RS D M . An R module is said to be free if it has a basis. Every vector space V over a field K is free as a K –module. (We have shown this for finite dimensional vector spaces, i.e., finitely generated K modules.) Modules over other rings need not be free. For example, any finite abelian group G is a Z –module, but no non-empty subset of G is linearly independent; in fact, if n is the order of G , and x 2 G , then nx D 0 , so f x g is linearly dependent. The R –module R n is free with the basis f O e 1 ; : : : ; O e n g , where O e j is the
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Unformatted text preview: sequence with j –entry equal to 1 and all other entries equal to . We call this the standard basis of R n . Proposition 8.1.28. Let M be an R –module and let x 1 ;:::;x n be distinct nonzero elements of M . The following conditions are equivalent: (a) The set B D f x 1 ;:::;x n g is a basis of M . (b) The map .r 1 ;:::;r n / 7! r 1 x 1 C r 2 x 2 C ±±± C r n x n is an R –module isomorphism from R n to M . (c) For each i , the map r 7! rx i is injective, and M D Rx 1 ˚ Rx 2 ˚ ±±± ˚ Rx n : Proof. It is easy to see that the map in (b) is an R –module homomorphism. The set B is linearly independent if, and only if, the map is injective, and B generates M if, and only if the map is surjective. This shows the equiv-alence of (a) and (b). We leave it as an exercise to show that (a) and (c) are equivalent. n...
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