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Free basis and basis?

  1. May 9, 2007 #1
    1. The problem statement, all variables and given/known data
    What are the differences between the two?
     
  2. jcsd
  3. May 10, 2007 #2
    Do you mean ordered basis?
     
  4. May 10, 2007 #3
    Don't think so.
     
  5. May 10, 2007 #4

    quasar987

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    Just surfing wiki, I get the impression that there is none. Do you have a reason to believe that there is a difference?
     
  6. May 10, 2007 #5
    Yeah, looks the same to me. This is what I studied in my linear algebra course: there's something called a basis and then something called an ordered basis. A basis with an ordering of elements (somewhat like an indexing set that has been fixed which maps to elements of your basis) is called an ordered basis.

    I think when you say free basis, you probably mean a basis without ordering...but thats just a wild guess. I haven't come across this term (free basis) in my reading of any standard textbook on Linear Algebra (cf Hoffman/Kunze).
     
  7. May 10, 2007 #6
    I am talking about it in the sense of modules.
     
  8. May 11, 2007 #7

    quasar987

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    If you look here

    http://en.wikipedia.org/wiki/Free_module

    they define a free module as a module having a free basis. But then they give the definition of a free basis and it is an exact analogy to what is simply called a 'basis' in linear algebra:

    http://en.wikipedia.org/wiki/Basis_(linear_algebra)#Definition

    Another reason why I suspect that there is no difference btw the two is the following sentence in the article about modules:

    "However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules,[...] "
     
    Last edited: May 11, 2007
  9. May 11, 2007 #8

    matt grime

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    Vector spaces are modules for fields. All modules for fields are free, so in this case the two notions agree.
     
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