Question: Suppose K and L are fields and . Show that L is a vector space over K.

Firstly, I'm not quite sure what it means to say "L is a vector space over K" and secondly, I'm not quite sure how to show that L is a vector space. Help please?

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- Sep 28th 2011, 01:22 PMworc3247Question about fields and vector spaces
Question: Suppose K and L are fields and . Show that L is a vector space over K.

Firstly, I'm not quite sure what it means to say "L is a vector space over K" and secondly, I'm not quite sure how to show that L is a vector space. Help please? - Sep 28th 2011, 02:03 PMChris11Re: Question about fields and vector spaces
Hint. The set of real numbers is a vectorspace over its self, can you find a subfield Kof R that lets (R, K) into a vector space? I'm sure you can. Use this idea in your general case.

- Sep 28th 2011, 02:43 PMDrexel28Re: Question about fields and vector spaces
- Sep 28th 2011, 04:17 PMHallsofIvyRe: Question about fields and vector spaces
- Sep 28th 2011, 04:32 PMDevenoRe: Question about fields and vector spaces
just in case you have forgotten, here is the definition:

a vector space over a field F is a set V, together with two operations: +:VxV--->V and *:FxV-->V such that:

1.(V,+) is an abelian group

2. for c in F, and u,v in V, c*(u+v) = c*u + c*v

3. for c,d in F, and u in V, (c+d)*u = c*u + d*u

4. for c,d in F, and u in V, c*(d*u) = (cd)*u

5. for u in V, and the (multiplicative) identity 1 of F, 1*u = u.

for a subfield K of a field L, we may take F = K and V = L (yes, L has additional structure, but that is beside the point).

the vector + operation is defined to be the addition of L.

the scalar multiplication is defined to be k*l = kl, the product of k and l in the field L.

now, verify the axioms 1 through 5.

(the normal example for this kind of thing is the complex field, and the real field, where the complex field is a 2 dimensional vector space over the reals, with basis {1,i}). - Sep 28th 2011, 05:30 PMymarRe: Question about fields and vector spaces
With or without definitions, one cannot solve an ambiguously stated problem. A vector space is a tuple of which V is only one element. Another element of this tuple is the multiplication of elements of the field K and elements of V. What this multiplication is supposed to be isn't said in the problem, although it is rather obvious. Actually, other operations of the space aren't defined either.

I don't think I've seen this problem formulated unambiguously though. - Sep 28th 2011, 05:48 PMDevenoRe: Question about fields and vector spaces
that is a good point. it is possible some other operation might be intended than just the (given) field multiplication in L. and the vector sum isn't explicitly stated, either.

a better way to phrase the question would be:

"Prove that if K is a subfield of a field L, that L is a vector space over K, with vector addtion being the field addition of L, and scalar multiplication being the field multiplication of L". (being well-defined since K is entirely contained within L).

because given even just an abelian group, it is possible to define other operations on that set and wind up with a different abelian group with the same underlying set.

that said, i don't think the ambiguity of the wording of the problem, is what is giving the original poster his difficulty. - Sep 28th 2011, 06:24 PMymarRe: Question about fields and vector spaces
I agree, although I can imagine it being troublesome. I've been there. It's not easy when you have to cope with completely new concepts defined in abstract ways. I still find myself spewing out profanity at the authors of ambiguous sentences in math books and later wondering how I could have any trouble with them. Sometimes an hour later. :)

- Sep 29th 2011, 06:10 AMworc3247Re: Question about fields and vector spaces
Ok thanks, I think I've got it.

- Sep 29th 2011, 06:26 AMSwlabrRe: Question about fields and vector spaces
- Sep 29th 2011, 06:27 AMworc3247Re: Question about fields and vector spaces
- Sep 29th 2011, 07:16 AMDevenoRe: Question about fields and vector spaces
- Sep 29th 2011, 07:26 AMymarRe: Question about fields and vector spaces