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?
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?
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.
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.
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.