Editing General position (section)

==General linear position==
A set of points in a {{mvar|d}}-[[dimension]]al [[affine space]] ({{mvar|d}}-dimensional [[Euclidean space]] is a common example) is in '''general linear position''' (or just '''general position''') if no {{mvar|k}} of them lie in a {{math|(''k'' &minus; 2)}}-[[dimension]]al [[Flat (geometry)|flat]] for {{math|1=''k'' = 2, 3, ..., ''d'' + 1}}. These conditions contain considerable redundancy since, if the condition holds for some value {{math|''k''<sub>0</sub>}} then it also must hold for all {{mvar|k}} with {{math|2 ≤ ''k'' ≤ ''k''<sub>0</sub>}}. Thus, for a set containing at least {{math|''d'' + 1}} points in {{mvar|d}}-dimensional affine space to be in general position, it suffices that no [[hyperplane]] contains more than {{mvar|d}} points – i.e. the points do not satisfy any more linear relations than they must.<ref>{{harvnb|Yale|1968|loc=p. 164}}</ref>

A set of at most {{math|''d'' + 1}} points in general linear position is also said to be ''affinely independent'' (this is the affine analog of [[linear independence]] of vectors, or more precisely of maximal rank), and {{math|''d'' + 1}} points in general linear position in affine ''d''-space are an [[affine basis]]. See [[affine transformation]] for more.

Similarly, ''n'' vectors in an ''n''-dimensional [[vector space]] are linearly independent [[if and only if]] the points they define in [[projective space]] (of dimension {{math|''n'' &minus; 1}}) are in general linear position.

If a set of points is not in general linear position, it is called a [[degenerate case]] or degenerate configuration, which implies that they satisfy a linear relation that need not always hold.

A fundamental application is that, in the plane, [[five points determine a conic]], as long as the points are in general linear position (no three are collinear).