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Topic: Basis of module and basis of vector space qs.
Replies: 9   Last Post: Nov 21, 2013 10:08 PM

 Messages: [ Previous | Next ]
 magidin@math.berkeley.edu Posts: 11,749 Registered: 12/4/04
Re: Basis of module and basis of vector space qs.
Posted: Nov 19, 2013 10:53 AM

On Tuesday, November 19, 2013 7:59:57 AM UTC-6, Sandy wrote:
> Arturo Magidin wrote:
>

> > On Monday, November 18, 2013 2:17:05 PM UTC-6, Sandy wrote:
>
> >> Is it the case that
>
> >>
>
> >>
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> >>
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> >> a module has a basis iff it is a free module?
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> >
>
> > Yes.
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> >
>
> >
>
> >> Is it the case that
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> >>
>
> >>
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> >>
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> >> (i) every vector space has a basis iff the axiom of choice holds;
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> >
>
> > Yes.
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> >
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> >
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> >> (ii) for every vector space V,
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> >>
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> >> (if B and C are bases of V then, there is a bijection: B -> C)
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>
>
> " then," was probably meant to be ", then" :-(.
>
>
>

> >> iff the axiom of choice holds
>
> >
>
> > I don't know for sure, but my dim memory is that this is strictly weaker than the full axiom of choice. You might want to check out the website/book on equivalents to the Axiom of Choice by the Rubins.
>
>
>
> Thank you. I know of the book but I don't have ready access to it, so I
>
> shall firtle around the Internet.

Try math.stackexchange.com; or perhaps mathoverlow.net

Date Subject Author
11/18/13 Sandy
11/19/13 magidin@math.berkeley.edu
11/19/13 Sandy
11/19/13 magidin@math.berkeley.edu
11/19/13 Sandy
11/19/13 magidin@math.berkeley.edu
11/19/13 Sandy
11/21/13 fom
11/21/13 Sandy
11/21/13 fom