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Math Help - proof

  1. #1
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    proof

    Let W be a subspace of a vector space V over a field F. For any v belonging to V the set {v} + W = {v+w: where w belongs to W} is called the coset of W containing v.

    Prove that v+W is a subspace of V iff v belongs to W.

    ??
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  2. #2
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    This is a simple idea.
    v \in W\quad  \Rightarrow \quad v + W = W.

    If v+W is a subspace then \left[ {\exists a \in W} \right]\left( {v + a = 0} \right).
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  3. #3
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    Quote Originally Posted by ruprotein View Post
    Prove that v+W is a subspace of V iff v belongs to W.
    Here is another exaplation.

    If you studied group theory....
    aH=H of a subgroup H or G and a\in G.
    If and only if a\in H.
    The reason is as follows, the relation that defines cosets is an equivalence relation, hence it divides cosets into disjoint sets, and the set that contains a is aH. Thus, because of disjointness we have that if aH=H then a\in H.

    Now by definition a vector space is a group (abelian) thus the same discussion in the preceding paragraph applies to a group as well.
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  4. #4
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    how does by saying v + a = 0 say that v belongs to W
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  5. #5
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    Isn't -a in W?
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  6. #6
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    Quote Originally Posted by ruprotein View Post
    how does by saying v + a = 0 say that v belongs to W
    Because,
    v+a=0
    Thus,
    v=-a
    But if a\in V then -a\in V (one of the axioms for a vector space).
    Thus,
    v\in V
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