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Vector spaces over fields other than R or C?

  1. Dec 28, 2006 #1
    Er that's it really. In the various texts I've got that introduce vector spaces, they always say 'defined over a field' and then give R and C as examples. Are there other fields that mathmos or physicists define vector spaces over?

    Does a vector space satisfy the axioms of a field? I'll think about it when I get home. If it did then you could have a vector space over vectors. :rofl:

    Oh and a quick Q about terminology. If we consider the arrows-in-3D-space that we all know and love from high school, do we say that's a 3-D vector space over R, or a vector space over R^3? Would the latter even make sense?


    I'm an accountant by the way. :uhh:
     
  2. jcsd
  3. Dec 28, 2006 #2

    matt grime

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    The next most common field is F_2, the field with 2 elements - it is the field that computers think in, to invent a white lie. The rationals, Q, are useful.

    No, vector spaces are not fields. They clearly don't satisfy the axioms of a field (out of the box, that is: there is no such thing as a good multiplication of vectors that makes sense in general, and make any vector space into a field).

    We say, 'three dimensional vector space over R', or 'three dimensional real vector space'.

    We do not say 'a vector space over R^3'. That does not make sense. R^3 is not a field.
     
  4. Dec 28, 2006 #3
    Cool, thanks for the comprehensive reply!

    So by 'next most common' I presume you mean 'frequently studied by mathematicians/computer scientists/etc'. But do they just study the fields themselves, or have interesting/useful vector spaces over them been studied?

    Hang on... would a byte (or indeed any length sequence of bits) then be a valid vector in a vector space over f_2? I think so but this is all quite new and scary.
     
  5. Dec 28, 2006 #4

    matt grime

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    No, I don't mean studied by mathematicians or computer scientists, per se. I wouldn't like to say what they do. It is however the next most natural field, i.e. one you'll find in real life situations, which is I presume how you're thinking of mathematics - by its uses.

    The answer as to whether a byte is a vector or not is a bit 'yes and no'. Sorry. A vector space over F_2 would look like a method of encoding bit strings. Whether or not you chose to do that is a different matter. It is useful to use the vector space ideas for, say, coding theory. I doubt it would help you produce a better processor, but then I don't make chips.

    I would say a byte is a unit of information. A vector is an element of a vector space (not arrows, please don't use arrows!).


    Remember, anything 'might' be a vector in some vector space. You need to specify what the space is, and that includes things like addition of vectors, multiplication by scalars and the zero vector, etc.
     
  6. Dec 28, 2006 #5

    mathwonk

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    yes, sounds right. any sequence of zeroes and 1's is vector ove Z/2.


    one advantge fields have over rings is you can divide, the main mybe only one rewlly.


    that makes it possible to reduce any spanning set to a basis, i,.e,. an in dependent spanning set.

    in trying to prove the decomposition theorem for finitely generated modules over a principal ideal domain like k[X] where k is a field, inability to do this is an obstackle.

    but in case the minimal polynomial for your operator, or your module, is "square free", i.e. a product of distinct irreducible polynomials, you can reduce to the case of a module over k[X]/(F) where f is irreducible, and this quotient ring is a field.

    that makes the proof easy for that special acse. i discoevred this for myself while writing out my webnotes on linear algebra a couple xamses ago. so i use fact that a module over the ring k[X] whose minimal polynomial f is irreducible, becomes a vector space over the field k[X]/(f). in my proof.
     
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