Editing Distance geometry (section)

==== Affine independence ====
Given the points <math display="inline">A_0, A_1,\ldots, A_n \in \mathbb R^k</math>, they are defined to be [[Affine independence|affinely independent]], [[iff]] they cannot fit inside a single <math>
l</math>-dimensional affine subspace of <math> \mathbb{R}^k</math>, for any <math> \ell < n</math>, iff the <math>n</math>''-''[[simplex]] they span, <math>v_n</math>, has positive <math>n</math>-volume, that is, <math>\operatorname{Vol}_n(v_n) > 0</math>.

In general, when <math>k\ge n </math>, they are affinely independent, since a [[Generic property|generic]] ''n''-simplex is nondegenerate. For example, 3 points in the plane, in general, are not collinear, because the triangle they span does not degenerate into a line segment. Similarly, 4 points in space, in general, are not coplanar, because the tetrahedron they span does not degenerate into a flat triangle.

When <math> n > k</math>, they must be affinely dependent. This can be seen by noting that any <math>n</math>-simplex that can fit inside <math>\mathbb{R}^k</math> must be "flat".