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Topic: What does one call vector geometry without a coordinate system?
Replies: 32   Last Post: Sep 9, 2013 4:45 PM

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 Shmuel (Seymour J.) Metz Posts: 3,473 Registered: 12/4/04
Re: What does one call vector geometry without a coordinate system?
Posted: Sep 2, 2013 9:38 AM

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).

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

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