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Thread: 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.
    $\displaystyle v \in W\quad \Rightarrow \quad v + W = W.$

    If v+W is a subspace then $\displaystyle \left[ {\exists a \in W} \right]\left( {v + a = 0} \right).$
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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....
    $\displaystyle aH=H$ of a subgroup $\displaystyle H$ or $\displaystyle G$ and $\displaystyle a\in G$.
    If and only if $\displaystyle a\in H$.
    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 $\displaystyle a$ is $\displaystyle aH$. Thus, because of disjointness we have that if $\displaystyle aH=H$ then $\displaystyle a\in H$.

    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,
    $\displaystyle v+a=0$
    Thus,
    $\displaystyle v=-a$
    But if $\displaystyle a\in V$ then $\displaystyle -a\in V$ (one of the axioms for a vector space).
    Thus,
    $\displaystyle v\in V$
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