
Re: What does one call vector geometry without a coordinate system?
Posted:
Sep 2, 2013 9:38 AM


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).
 Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
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