linear function
Let ๐ฎ1=(๐ซ1,โ1) and ๐ฎ2=(๐ซ2,โ2) be two near-linear spaces.
Definition. A linear function from ๐ฎ1 to ๐ฎ2 is a mapping on the points that sends lines of ๐ฎ1 to lines of ๐ฎ2. In other words, a linear function is a function ฯ:๐ซ1โ๐ซ2 such that
ฯ(โ)โโ2 for every โโโ1. |
Here, ฯ(โ) is the set {ฯ(P)โฃPโโ}. A linear function is also called a homomorphism.
When both ๐ฎ1 and ๐ฎ2 are linear spaces, then ฯ being a linear function is equvalent to saying that P,Q are collinear
iff ฯ(P),ฯ(Q) are collinear.
If ๐ฎ1 is a linear space, then so is (ฯ(๐ซ1),ฯ(โ1)). This shows that if ฯ:๐ฎ1โ๐ฎ1 is onto, ๐ฎ2 is a linear space if ๐ฎ1 is.
Let ฯ:๐ฎ1โ๐ฎ2 be a one-to-one linear function. If points P1โ P2 lie on line โ, then ฯ(P1)โ ฯ(P2) lie on ฯ(โ). This also shows that three collinear points in ๐ฎ1 are mapped to three collinear points in ๐ฎ2. In addition, we have
|โ|=|ฯ(โ)| for any line โ in ๐ฎ1.
Definition. When ฯ:๐ฎ1โ๐ฎ2 is a bijection whose inverse
ฯ-1 is also linear, we say that ฯ is an isomorphism
. When ๐ฎ1=๐ฎ2=๐ฎ, we call ฯ an automorphism, or more commonly among geometers, a collineation
, of the space ๐ฎ.
Suppose ฯ:๐ฎ1โ๐ฎ2 is an isomorphism. For every point P, let P* be the set of all lines passing through P. Then
|P*|=|ฯ(P)*| for any point P in ๐ฎ1.
It is possible to have a bijective linear function whose inverse is not linear. For example, let ๐ฎ1 be the space with two points P,Q with no lines, and ๐ฎ2 the space with the same two points with line {P,Q}. Then the identity function on {P,Q} is a bijective linear function whose inverse is not linear. On the other hand, if the both spaces are linear, then the inverse is always linear.
Remark. The usage of the term โlinear functionโ differs from its more usual meaning as a linear transformation between vector spaces in the study of linear algebra.
References
- 1 L. M. Batten, Combinatorics of Finite Geometries, 2nd edition, Cambridge University Press (1997)
Title | linear function |
---|---|
Canonical name | LinearFunction |
Date of creation | 2013-03-22 19:14:46 |
Last modified on | 2013-03-22 19:14:46 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 12 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 51A45 |
Classification | msc 51A05 |
Classification | msc 05C65 |