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.