Talk:Collinearity: Difference between revisions

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The net is redirecting the searched contents to a more basic meaning. 141.214.17.17 (talk) 21:08, 7 August 2008 (UTC)[reply]

Colinear maps have nothing im common with collinearity. It is a mere syntactic resemblence. --Tillmo (talk) 16:10, 2 March 2009 (UTC)[reply]

How is this relevant to a largely-mathematical concept? Just saying "Collinearity (also co-linearity and colinearity)" would be sufficient, if you REALLY wanted people to know about the alternative spellings. —Preceding unsigned comment added by 72.248.107.194 (talk) 19:55, 1 May 2009 (UTC)[reply]

The above comment is a good example of why more needs to be said than just common misspellings. There are many readers who think that the one "l" version of the term is an alternate spelling and are unaware that that spelling changes the meaning. In this day and age, when correct spelling is not considered a high priority, it is important to point out changes in meaning that can occur in this way. Perhaps the phrasing could be improved, but I definitely think that something needs to be said. As to the redundancy ... a little bit of it doesn't hurt the article, especially considering that it is a disambiguation page with readers coming to it with different mindsets as to what the term means. Furthermore, even if this was some terrible violation of Wiki "rules", the wrong one was eliminated. Bill Cherowitzo (talk) 05:34, 7 February 2012 (UTC)[reply]

I too thought (until a moment ago) that colinear is a misspelling, and I'm not aware of any different meaning of colinear. I just looked it up in Wiktionary [1], and found that Wiktionary claims that colinear is an alternative spelling of collinear, with no separate definition given. Random House Webster's College Dictionary agrees.

Do you have evidence that some authority considers colinear to be an incorrect spelling? And, what changed meaning do you say is attached to colinear?

As for the redundancy, there's simply no reason to say the same thing twice in a row; there's simply no reason to say the same thing twice in a row--it makes the disambiguation page look sloppy. And it is standard practice on disambiguation pages to put the primary usage in the opening sentence, separating it from the less common uses; so I think I deleted the right one. Duoduoduo (talk) 17:16, 7 February 2012 (UTC)[reply]

The word "colinear" refers to the dual of linear in certain algebraic contexts, as is pointed out on this disambiguation page. Your finding the term in general dictionarys just points out the problem that mathematical terms are just not treated accurately in those venues (and there is absolutely no reason to trust Wiktionary or any similar on-line source). Asking for an authority for an incorrect spelling is a red herring, how many authorities can you find that will say that "fisch" is a misspelling of the aquatic animal. There are no geometry texts that spell collinear with one "l" that I am aware of, I consider that to be the real authority on this issue. Bill Cherowitzo (talk) 23:19, 7 February 2012 (UTC)[reply]

And on the redundancy issue, while I certainly agree that a primary usage ought to appear in an opening sentence, there are no rigid rules about that ... and for emphasis I repeat, THERE ARE NO RIGID RULES ABOUT THAT! Even if there were, I would stand by WP:IAR and claim that the page is actually better off (clearer and less awkward) with this little bit of redundancy. Bill Cherowitzo (talk) 05:10, 8 February 2012 (UTC)[reply]

I've pulled the following sentence from the lead,

The term collinear has also been used elsewhere in mathematics and in its application areas, as a reference to concepts of line and linear dependence.

because I don't consider it to be correct (or at the least misleading). The term is not referring to a property of lines, nor an intrinsic property of points - it is a property of flags (the incidence relation). To refer just to lines (or just to points) is only telling half of the story. Linear dependence is an algebraic property. In order to bring collinearity into the picture you must be interpreting the situation in geometric terms. In general, this is not a natural way to view the algebraic relation. I don't remember seeing the term raised in any discussion of linear dependence. Of course some good references may correct this impression, but it would still be a stretch and I am not sure that it belongs in the lead. Perhaps if there were some mention of the connection in the article this could be justified. Bill Cherowitzo (talk) 18:52, 6 May 2013 (UTC)[reply]

It's likely that I'm just confused about this, but the following section seems inaccurate:

In coordinate geometry, in n-dimensional space, a set of three or more distinct points are collinear if and only if, the matrix of the coordinates of these vectors is of rank 1 or less. For example, given three points X = (x₁, x₂, ... , x_n), Y = (y₁, y₂, ... , y_n), and Z = (z₁, z₂, ... , z_n), if the matrix

${\displaystyle {\begin{bmatrix}x_{1}&x_{2}&\dots &x_{n}\\y_{1}&y_{2}&\dots &y_{n}\\z_{1}&z_{2}&\dots &z_{n}\end{bmatrix}}}$

is of rank 1 or less, the points are collinear.

IIRC, rank is just a measure of linear independence. But points need not be linearly dependent to be collinear. For example, the following points are all on the 2D line y = 1, but their coordinates are linearly independent: (1, 1), (2, 1), (3, 1). The matrix representation is:

${\displaystyle {\begin{bmatrix}1&1\\2&1\\3&1\end{bmatrix}}}$

which is of rank 2. I suspect that rank is actually a measure of collinearity only for lines that run through the coordinate origin.

The determinant formulation seems to be accurate.

NillaGoon (talk) 22:10, 4 June 2020 (UTC)[reply]

I agree with @NillaGoon, the following section seems incorrect:

In coordinate geometry, in n-dimensional space, a set of three or more distinct points are collinear if and only if, the matrix of the coordinates of these vectors is of rank 1 or less. For example, given three points X = (x₁, x₂, ... , x_n), Y = (y₁, y₂, ... , y_n), and Z = (z₁, z₂, ... , z_n), if the matrix

${\displaystyle {\begin{bmatrix}x_{1}&x_{2}&\dots &x_{n}\\y_{1}&y_{2}&\dots &y_{n}\\z_{1}&z_{2}&\dots &z_{n}\end{bmatrix}}}$

is of rank 1 or less, the points are collinear.

A counter-example is

${\displaystyle {\begin{bmatrix}1&2&3\\2&3&4\\3&4&5\end{bmatrix}}}$

This 3-point matrix is of rank 2, so according to the above claim, the points are not collinear.

But obviously,the points lie on a straight line, which brings a contradiction. Xfontaine (talk) 10:21, 16 October 2024 (UTC)[reply]

Yes, this is true. What you could do is subtract one of the columns from every column, and then if the resulting matrix is of rank 1 then the points represented by the columns are collinear. –jacobolus (t) 15:19, 16 October 2024 (UTC)[reply]

There are two errors here: firstly 1 must be replaced with 2. Then the assertion becomes true for projective coordinates only; for Cartesian coordinates and affine coordinates, a column of 1 must be added to the matrix. D.Lazard (talk) 17:14, 16 October 2024 (UTC)[reply]

Do we have references supporting these properties?

The Japanese page would require modification as well, since it is a translation of the present (English) one. The French page has issues as well. Xfontaine (talk) 03:13, 17 October 2024 (UTC)[reply]

Projective coordinates for points in n-dimensional projective space represent each point by an n+1-dimensional ratio (e.g. for a point in the projective plane, a point might be represented by the ratio 3:2:1, which is proportional to the ratio 6:4:2, etc.), so that if we take the constituents of every proportional ratio to represent coordinates in an n+1-dimensional affine space, the set of such coordinates describes a line through the origin; we can pick any point along that line other than the origin itself to be a vector representative of its direction.

If we make a matrix of such vectors (either as rows or columns), the rank of the matrix represents the dimension of the subspace spanned by those affine lines through the origin, which is 1 dimension higher than that of the projective subspace spanned by the original projective points, so that collinear points in our projective space would give a matrix of rank 2. I imagine any source about projective geometry from a coordinate point of view could be used as a citation for this basic fact. –jacobolus (t) 06:51, 17 October 2024 (UTC)[reply]

"Do we have references supporting these properties?" I have no explicit reference, but this can be found in most textbooks that contain the standard geometric interpretation of linear algebra. In fact, the result is a particular case of the more general result that the points of the subspace spanned by a given set of points are obtained by linear combinations, and therefore the dimension of the subspace spanned by a set of points is one less than the rank of the matrix of coordinates (augmented by a column of 1 in the affine case). The "one less" results in the projective case from the fact that scaling the coordinates with the same factor does not change the point. In the affine case the "one less" results from that one must consider only the linear combinations with 1 in the last column. D.Lazard (talk) 08:46, 17 October 2024 (UTC)[reply]

A curious new section.

• Two parallel lines are not necessarily colinear (do not necessarily coincide).

Not necessarily? What's an example of parallel lines that do coincide?

• A line segment can be extended infinitely into a line (containing the segment).

Trivially true, in contrast with the rest of the section. —Tamfang (talk) 04:41, 21 October 2024 (UTC)[reply]