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Question about zero element in vector spaces

  1. Apr 29, 2005 #1
    Suppose I have a set involving trigonometric functions, with addition defined as multiplication of two vectors. If this is a vector space, the zero vector has to be unique. If cos (0) works as the zero vector, then cos (2*pi), etc. also work. Does this mean the set is not a vector space, because the zero element is not unique? Or is it still a vector space (all other axioms check out) because cos (0) = cos (2*pi) = 1 ?
  2. jcsd
  3. Apr 29, 2005 #2
    I'm a little confused... what's your underlying field? How is multiplication between a field element and vector element defined?

  4. Apr 29, 2005 #3


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    As long as it satifies the axioms of a vector space it is a vector space. The set of vectors may be a partition of another set defined by an equiavlence relation (which I think is what you're getting at).
  5. Apr 30, 2005 #4

    matt grime

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    The zero vector in a vector space of functions is the ZERO function if f(x)=0 for all x. Is that the kind of thing you're after?
  6. Apr 30, 2005 #5
    I understand now. You guys were a lot of help!
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