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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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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08