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 equivalence of (a) and (b). We leave it as an exercise to show that (a) and (c) are equivalent. n...
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Vector Space, linearly independent set, Rn xn

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