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Determining if certain sets are vector spaces

  1. Dec 13, 2011 #1
    1. The problem statement, all variables and given/known data

    The set of all pairs of real numbers of the form (1,x) with the operations:
    (1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx) k being a scalar

    Is this a vector space?


    2. Relevant equations
    (1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx)


    3. The attempt at a solution
    I verified most of the axioms hold, but I'm unsure about the additive identity.
    Can it be something other than the zero vector?
    My attempt, "O" being the additive identity
    O=(1,0)
    A+O=(1,x)+(1,0)=(1,x)
    This means the additive inverse must equal (1,0)
    A+(-A)=(1,x)+(-1,-x)=(1,x+(-x))=(1,0)
    If this isn't right, then I know it doesn't hold. I'm just a little confused. Thanks for any help
     
  2. jcsd
  3. Dec 13, 2011 #2

    Dick

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    That's all correct. You understand this very well. They defined the operations in a way that was maybe intended to confuse you. But you weren't confused. (1,0) is the 'zero' vector.
     
  4. Dec 13, 2011 #3

    Office_Shredder

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    If A=(1,x) then -A is (1,-x) not (-1,-x). Otherwise it looks like you've got it. You can imagine just chopping off the 1 here and describing the element (1,x) by just the number x. Then what you really have is just the standard real numbers
     
  5. Dec 13, 2011 #4

    Dick

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    Ooops. Missed that. Thanks for being careful.
     
  6. Dec 13, 2011 #5

    SammyS

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    Sort of OK. Besides the correction above, the wording needs a little work.

    The sentence "This means the additive inverse must equal (1,0) ." should be changed to something like: "This means when the additive inverse of any number pair is added to that number pair, the result is (1,0)."
     
  7. Dec 13, 2011 #6
    Thank you all for the very quick responses!! That makes me feel so much better... I have a final coming up on this :)
     
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