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Homework Help: Prove that the additive identity in a vector space is unique

  1. Jan 16, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that the additive identity in a vector space is unique


    2. Relevant equations

    Additive identity

    There is an element 0 in V such that v + 0 = v for all v in V

    3. The attempt at a solution

    Assume that the additive identity is NOT unique, then there exists y, z belong to V such that
    A + y = A + z = A, then y = z = 0, which is a contradiction.

    Is this enough to prove??
     
  2. jcsd
  3. Jan 16, 2010 #2

    tiny-tim

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    Hi zeion! :smile:

    hmm … you're assuming that A - A = 0, which is sort-of begging the question.

    Hint: what is y + z ? :wink:
     
  4. Jan 16, 2010 #3
    Since y, z belong to V, and y, z are the zero vectors in V, then
    y + z = y = z, which is a contradiction.
     
  5. Jan 16, 2010 #4

    tiny-tim

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    Yup! :biggrin:

    (except it's not actually a contradiction … unless you state at the start that y and z are different, which you don't have to).
     
  6. Jan 16, 2010 #5
    Ooh okay! Thanks ^_^

    But why couldn't I say that A - A = 0?
    Could I do that if I stated that I assumed A was in V?

    ..or is it because then I would be assuming that A was unique?
     
  7. Jan 17, 2010 #6

    tiny-tim

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    … that 0 was unique? yes! :wink:
     
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