Date: Sep 2, 2013 10:21 AM
Author: fom
Subject: Re: What does one call vector geometry without a coordinate system?

On 9/2/2013 8:38 AM, Shmuel (Seymour J.) Metz wrote:
> In <65SdnZdjTuK-7b7PnZ2dnUVZ_gudnZ2d@giganews.com>, on 09/01/2013
> at 11:49 AM, fom <fomJUNK@nyms.net> said:
>

>> Could you explain your second statement a
>> little more.

>
> Affine Geometry is characterized by a general linear group and a
> translation group; there is no subgroup of GL(n) singled out. You can
> unambiguously define segments on parallel lines to be equal if there
> is a translation taking one into the other, but if the lines are not
> parallel then you need a transformation from GL(n), and it is too big
> for uniqueness.
>

>> Does it relate to the difference between a vector space and an inner
>> product space?

>
> Yes; a positive define inner product on V lets you single out a
> subgroup of V's symmetry group, e.g., O(n) c GL(n).
>


Thanks.

My general background had been in pure mathematics
and the role for the classical groups in relation
to geometric situations had never been explained
clearly.