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Math Help - Congruence modulo and equivalence classes

  1. #1
    Member Mollier's Avatar
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    Congruence modulo and equivalence classes

    Hi,

    problem: Suppose that M is a subspace of a vector space V. Prove that a subset of V is an equivalence class modulo M if and only if it is a coset of M

    attempt:
    Let N be a subset of V.
    1st direction: If N is a coset of M, then it is an equivalence class modulo M.
    N=x+M.
    Now, I don't really know how to go from N beeing a coset of M to it beeing an equivalence class modulo M. Any help is greatly appreciated!

    Thanks.
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  2. #2
    Junior Member nimon's Avatar
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    Hi,

    The only thing you can really do is show that

    N = x+M = [x] = \{y\in V : y-x \in M\}.

    This isn't too hard, you just need to keep the definitions in mind:

    y \in [x] \Leftrightarrow y - x \in M \Leftrightarrow y \in x + M = N.

    Just make sure you keep appealing to the definitions, and make sure you use the closure of M under addition and taking inverses (owing to the fact that it is a subspace) to convince yourself that x\sim y \Leftrightarrow x-y\in M actually does define an equivalence relation.
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