fom
Posts:
1,968
Registered:
12/4/12


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


On 9/2/2013 8:38 AM, Shmuel (Seymour J.) Metz wrote: > In <65SdnZdjTuK7b7PnZ2dnUVZ_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.

