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Identifying elements in V(x)W, V,W V.Spaces

  1. Dec 18, 2007 #1

    WWGD

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    Hi, everyone:
    I have been playing around with tensor products of vector spaces recently.
    This question came up:

    How can we tell if c(a(x)b)~(a(x)b) , or wether any two general elements
    are in the same class, other than in the std. cases like c(a(x)b)~(ca(x)v)~
    (a(x)cv) , etc.

    Any Ideas?.
    Thanks.
     
  2. jcsd
  3. Dec 18, 2007 #2

    mathwonk

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    this is why tensor products are hard, it is hard to tell when two reporesentatives for elements represent the same element of a tensor product.

    as a tiny example, show that every pair of representatives in the tensor product of Z/(37) and Z/(101) represent the same element. e.g. for all pairs of integers (n,m) and (a,b), their tensor products are equal in that group.
     
  4. Dec 20, 2007 #3

    WWGD

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    Thanks, Mathwonk. I hope I am not missing something obvious
    here, but I am kind of confused, tho; are you seeing Z/(37) and
    Z/(101) as fields?. How do you define the tensor product, then?
     
  5. Dec 21, 2007 #4

    WWGD

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    Sorry for my dumbness here, Mathwonk. I was stuck with the idea of tensoring
    vector spaces, and was thinking of both Z/(37) and Z/(101) as vector
    spaces over themselves , so that the tensor did not make sense. Then
    I thought for a second and realized it is a Z-module tensor.I'll do the exercise--
    and review my algebra.
     
  6. Dec 21, 2007 #5

    mathwonk

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    the case of vector spaces is trivial. the dimension of VtensW is always the product of their dimensions, and if {vi} and {wj} are bases, then so is {vitenswj}.
    so every element can be written in terms of the basis to see if two elements are equivalent. the hard cases are modules that are not free.
     
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