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For example, in space three-dimensional affine space generated by two skew lines is all the space three-dimensional, since they are not coplanar. For this reason it is not worth the Grassmann formula, which in this case would say that the space generated by the two straight lines has dimension 1 +1-0.

The affine geometry is intermediate between the geometry of vector spaces and the projective subspaces in a vector space are forced to pass through the origin. The affine space is then built in order to remedy this lack unnatural, but in doing so you lose the Grassmann formula, and many problems will lengthen the list of cases to consider: two straight lines can be accidents, coplanar, skew ... The projective space eliminates re-adding phenomena of parallelism of "new points at infinity", without restoring a "vantage point", and so here is Grassmann.

Could you help me understand this concept? In a strict and intuitive way.

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    $\begingroup$ Beside the point, but 'Why in the affine space' is really an amazing substitute for 'Why on earth'. $\endgroup$ – Abel May 22 '13 at 11:38
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The dimension will be the same in affine and projective geometry if you use the same definition of "generated", except that additional general position assumptions could be needed in the affine case. If you use different notions of generation for the two geometries then of course the answer could vary.

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