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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Question: Suppose K and L are fields and
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?
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.
What it is asking you to prove that if you define the multiplication of an element ofQuote:
Originally Posted by worc3247
Question: Suppose K and L are fields and
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?
and an element of
in the normal way (the way the field says so) then this makes
You can't do any math problems if you don't know the definitions! What is the definition of "V is a vector space over field K"?Quote:
Originally Posted by worc3247Question: Suppose K and L are fields and
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?
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}).
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.Quote:
Originally Posted by HallsofIvy
I don't think I've seen this problem formulated unambiguously though.
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.Quote:
Originally Posted by ymar
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.
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. :)Quote:
Originally Posted by Deveno
Ok thanks, I think I've got it.
To quote HallsOfIvy, "You can't do any math problems if you don't know the definitions!"Quote:
Originally Posted by worc3247
so, what is the definition of a Field?
(My point is that all fields have scalar multiplication!)
Yes I got it now :) thanks!Quote:
Originally Posted by Swlabr
well, the "field" with 1 element (the so called "F-un") doesn'tQuote:
Originally Posted by Swlabr
(note to original poster, and others: this is a mathematical in-joke, there is no field with 1 element. but if there were....)
Interestingly, it was Tits who contributed to the discussion of F-un.Quote:
Originally Posted by Deveno