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Math Help - Prove that the additive inverse of a vector is unique

  1. #1
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    Prove that the additive inverse of a vector is unique

    Suppose V is a vector space over a field F. Prove that the additive inverse of a vector v\in V is unique.

    A hint provided for the question is as follows:
    Suppose x,y\in V are additive inverses of v. Prove that x=y using the axioms of a vector space.

    My work so far (uses the hint):
    Suppose \exists v\in V such that x, y\in V are inverses of v.
    Then x=x+(v+y)=x+0=(x+v)+y=0+y=y.
    I'm pretty sure that my current answer is lacking substance, so I could use a hand in making it more solid.

    The main issue I'm having is some of the wording. I feel as though there's something I could be missing in the question. The part on "axioms of a vector space" is one I don't entirely understand.
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  2. #2
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    Quote Originally Posted by Runty View Post
    Suppose V is a vector space over a field F. Prove that the additive inverse of a vector v\in V is unique.

    A hint provided for the question is as follows:
    Suppose x,y\in V are additive inverses of v. Prove that x=y using the axioms of a vector space.

    My work so far (uses the hint):
    Suppose \exists v\in V such that x, y\in V are inverses of v.
    Then x=x+(v+y)=x+0=(x+v)+y=0+y=y.
    I'm pretty sure that my current answer is lacking substance, so I could use a hand in making it more solid.

    The main issue I'm having is some of the wording. I feel as though there's something I could be missing in the question. The part on "axioms of a vector space" is one I don't entirely understand.
    Looks fine to me except I would switch your second and third expressions to get

    x=x+0=x+(v+y)=(x+v)+y=0+y=y

    You used associativity of addition, which is an axiom.

    (Note: Since we never used commutativity, this proof also holds for non-abelian groups.)
    Last edited by undefined; September 16th 2010 at 06:41 AM. Reason: improved wording, added additional note
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