Talk:General position

Incomplete Sentence

This sentence does not make sense: A set of 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 d+1 points in general linear position in affine d-space are an affine basis.

If you take out the parenthetical remark it reads: A set of points in general linear position is also said to be affinely independent, and d+1 points in general linear position in affine d-space are an affine basis.

Since it is defining affine independence it should read something like: A set of points in general linear position is also said to be affinely independent if blah-blah-whatever-the-actual-definition-is.

Maybe it should read this: A set of points in general linear position is also said to be affinely independent if no one point is an affine combination of the others. — Preceding unsigned comment added by Quadelirus (talk • contribs) 01:18, 6 May 2012 (UTC)

Quadelirus (talk) —Preceding undated comment added 01:15, 6 May 2012 (UTC).

Referenced textbook said something different to what is said in the article

" In more generality, a set containing {\displaystyle k} k points, for arbitrary {\displaystyle k} k, is in general linear position if and only if no {\displaystyle (k-2)} (k-2)-dimensional flat contains all {\displaystyle k} k points.[1]"

This sentence says nothing of the dimensionality of the space occupied by the k points, which I think it critical. — Preceding unsigned comment added by 122.107.209.169 (talk) 11:02, 29 May 2017 (UTC)

It is not critical. Four points are in general position provided no plane (2-dimensional flat) contains all four points. This is true no matter what the dimension of the ambient space is; 5, 10, 100 it makes no difference. This is still true in a 2-dimensional space, but no four points can ever be in general position in that setting. Ouch! --Bill Cherowitzo (talk) 16:05, 29 May 2017 (UTC)

This gives contradicting definitions. The preceding definition says that four points in 2-dimensional space, none of which are on the same line, are in general position. Mattb42 (talk) 18:49, 8 November 2017 (UTC)

I'm not sure of what you are saying. In a 2-dimensional space, any two distinct points lie on (determine) a line, so the phrase "none of which are on the same line" does not make sense to me. I am also not sure about which definitions seem to be in conflict. According to the definition at the start of this thread, 4 points are in general position provided no 2(= 4 − 2)-dimensional subspace contains them all. Since they all live in a 2-dimensional space (and a space is a subspace of itself) by assumption, they can not be in general position. But this definition is incorrect --Bill Cherowitzo (talk) 19:11, 8 November 2017 (UTC)

Sorry, I meant no 3 of which lie on the same line. The preceding definition: "A set of at least d+1 points in d-dimensional affine space is said to be in general linear position if no hyperplane contains more than d points" implies that a set of 4 points in 2-dimensional space are in general linear position if no line contains more than 2 points. Mattb42 (talk) 19:22, 8 November 2017 (UTC)

Okay, you are right. Three or more points in a plane are in general position iff no three are collinear (which I am embarrassed to say I know very well). Looking back at the history of this page I see that there have been many changes, and some conditions have been dropped over time. I will check the reference (I have a copy) and fix this tomorrow. --Bill Cherowitzo (talk) 04:59, 9 November 2017 (UTC)

I have replaced the definition by what actually appears in the source. Back in 2010 someone had removed the limit on k which meant that the definitions were no longer consistent (one was supposed to be just a special case of the other). --Bill Cherowitzo (talk) 20:05, 9 November 2017 (UTC)