|
|
| Line 1: |
Line 1: |
| A '''linear space''' is a basic structure in [[incidence geometry]]. A linear space consists of a set of elements called '''points''', and a set of elements called '''lines'''. Each line is a distinct subset of the points. The points in a line are said to be '''incident''' with the line. Any two lines may have no more than one point in common. Intuitively, this rule can be visualized as two straight lines, which never intersect more than once.
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. To climb is something I truly enjoy performing. For many years he's been living in Alaska and he doesn't plan on changing it. For many years she's been working as a journey agent.<br><br>Look into my website ... good psychic ([http://www.prograd.uff.br/novo/facts-about-growing-greater-organic-garden navigate to this site]) |
| | |
| ''Linear spaces'' can be seen as a generalization of [[projective plane|projective]] and [[affine plane (incidence geometry)|affine planes]], and more broadly, of [[block design|2-<math>(v,k,1)</math> block designs]], where the requirement that every block contains the same number of points is dropped and the essential structural characteristic is that 2 points are incident with exactly 1 line.
| |
| | |
| The term ''linear space'' was coined by [[Libois]] in 1964, though many results about linear spaces are much older.
| |
| | |
| ==Definition==
| |
| Let ''L'' = (''P'', ''G'', ''I'') be an [[incidence structure]], for which the elements of ''P'' are called points and the elements of ''G'' are called lines. ''L'' is a ''linear space'' if the following three axioms hold:
| |
| | |
| *(L1) two points are incident with exactly one line.
| |
| *(L2) every line is incident to at least two points.
| |
| *(L3) ''L'' contains at least two lines.
| |
| | |
| Some authors drop (L3) when defining linear spaces. In such a situation the linear spaces complying to (L3) are considered as ''nontrivial'' and those who don't as ''trivial''.
| |
| | |
| ==Examples==
| |
| The regular [[Euclidean plane]] with its points and lines constitutes a linear space, moreover all affine and projective spaces are linear spaces as well.
| |
| | |
| The table below shows all possible nontrivial linear spaces of five points. Because any two points are always incident with one line, the lines being incident with only two points are not drawn, by convention. The trivial case is simply a line through five points.
| |
| | |
| In the first illustration, the ten lines connecting the ten pairs of points are not drawn. In the second illustration, seven lines connecting seven pairs of points are not drawn.
| |
| | |
| {| class="wikitable" align="center"
| |
| |-
| |
| | style="width:150px" valign="center" align="center" | [[Image:Linear space1.png]]
| |
| | style="width:150px" valign="center" align="center" | [[Image:Linear space2.png]]
| |
| | style="width:150px" valign="center" align="center" | [[Image:Linear space3.png]]
| |
| | style="width:150px" valign="center" align="center" | [[Image:Linear space4.png]]
| |
| |-
| |
| | align="center" | 10 lines
| |
| | align="center" | 8 lines
| |
| | align="center" | 6 lines
| |
| | align="center" | 5 lines
| |
| |}
| |
| | |
| A linear space of ''n'' points containg a line being incident with ''n'' − 1 points is called a ''near pencil''. (See [[Pencil (mathematics)|pencil]])
| |
| | |
| {| class="wikitable" align="center"
| |
| |-
| |
| |style="width:250px" valign="center" align="center" | [[Image:Linear space near pencil.png]]
| |
| |-
| |
| | align="center" | near pencil with 10 points
| |
| |}
| |
| | |
| ==See also==
| |
| *[[Block design]]
| |
| *[[Fano plane]]
| |
| *[[Molecular geometry]]
| |
| *[[Partial linear space]]
| |
| | |
| ==References==
| |
| *A. Beutelspacher: Einführung in die endliche Geometrie II. Page 159, Bibliographisches Institut 1983, ISBN 3-411-01648-5 (German)
| |
| * J.H. van Lint, R.M. Wilson: A Course in Combinatorics. Page 188, Cambridge University Press 1992,ISBN 0-521-42260-4
| |
| * L.M. Batten, A. Beutelspacher: The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, 1992.
| |
| | |
| [[Category:Incidence geometry]]
| |
I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. To climb is something I truly enjoy performing. For many years he's been living in Alaska and he doesn't plan on changing it. For many years she's been working as a journey agent.
Look into my website ... good psychic (navigate to this site)