Vector spaces over fields other than R or C?

[Nancarrow]

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.

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.

 

[matt grime]

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.

 

[Nancarrow]

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.

 

[matt grime]

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.

 

[mathwonk]

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.