Re: Vector spaces over real numbers

On Wed, 03 Oct 2007 20:15:12 +0000, Daniel Kraft <d@xxxxxxxx> wrote:

Virgil wrote:
In article <fe0a7f$iuj$1@xxxxxxxxxxxxxxxxxxxxx>,
Daniel Kraft <d@xxxxxxxx> wrote:

Hi,

here are two questions which turned up solving some "homework" problems
for a basic linear algebra course:

1) Are the rational numbers a vector space over the real numbers?

More interesting in a number of ways is the issue of the reals as a
vector space over the rationals.

Well, this was the problem statement; and as the rationals are no vector
space over the reals, it's of course not quite "interesting" ;)

What do you think are the interesting properties of reals over rationals?

Cheers,
Daniel

Here's something to look up - the properties of a module, which a
vector space is, but a module is a much more general version of such a
thing.

Here are the properties of a module, and then you can see where the
reals, as a vector space over the rationals, would fit in.

"Let R be a ring. A left R-module is an abelian group M, written
additively, on which R acts linearly."

That's the definition given in Farb and Dennis' "Noncommutative
Algebra". Even those authors realize that such a definition is cool,
but it can be supplemented with more information.

So, a left R-module M is a nonzero set with a map, RxM --> M, where
(r,m) is denoted by r m with the given properties

(r + s) m = r m + sm, for every r,s in R, every m in M
r(m + n) = r m + r n, for every r in R, every m,n in M
(rs) m = r (sm), for every r,s in R, every m in M
1 m = m, for every m in M.

Obviously, this assumes that R has a multiplicative identity. If you
look at the vector space axioms, this is clearly a generalization.
Things like uniqueness of vector representation, etc. can be realized
via things like Free modules, and semisimple modules.

Personally, I was interested in exactly what tweakings of the ring and
module that would produce that uniqueness of representation of
vectors. In module theory, they simply call the modules Free modules.
But, given a module, what are the exact additions that can be made to
the ring it to make the module a vector space?

Let me explain, if the ring R is a field or division ring, the module
is a vector space. However, suppose a module takes exactly n
properties to become a vector space, then what property is n-1? Or,
what are the n-1 properties that need to be there right before the
module becomes a vector space?

Similarly, given a ring R, suppose there are m properties to make the
module into a vector space - I know the ring will have to have some
minimal properties, but just suppose those are true for the moment. If
it takes m properties to make the module a vector space, then what are
the m-1 properties?

I'm looking for the exact "moment" a module becomes a vector space, if
that makes sense. There is, I think, a balance of properties between
the ring and the module that work together to make a vector space - at
least in the way we think in terms of unique representation of a
vector as a linear combination of other vectors in the module.

This might have an answer, although I have never seen one.

Thanks,

Brian

.

Relevant Pages

  • Re: fourrier-budan
    ... a field is a vector space over any subfield. ... The dimension of R as a vector field over R is 1, ... ring containing R and z. ... fractional dimension as a vector space over the reals. ...
    (sci.math)
  • Re: Complexes of a Group?
    ... term `module' as a synonym for Abelian group. ... is defined by specifying a ring, and an abelian group. ... they, in fact, do form a one dimensional vector space. ... I am interested in valid categorizations of the integers, rationals, reals, ...
    (sci.math)
  • Re: Vector spaces over real numbers
    ... Are the rational numbers a vector space over the real numbers? ... scalar multiplication, yes, you are correct. ... way to define scalar multiplication in order to make the rationals ... specify/postulate anything like this for vector spaces over the reals... ...
    (sci.math)
  • Re: How to define a vector without coordinate system?
    ... The real numbers are a vector space OVER THE RATIONALS. ... Sqrt 2 is a some series of rational numbers. ... thus an explicit "coordinate system" for that particular vector space, ...
    (sci.math)
  • Re: Vector spaces over real numbers
    ... for a basic linear algebra course: ... Are the rational numbers a vector space over the real numbers? ... More interesting in a number of ways is the issue of the reals as a ... vector space over the rationals. ...
    (sci.math)