linear function
Let and be two near-linear spaces.
Definition. A linear function from to is a mapping on the points that sends lines of to lines of . In other words, a linear function is a function such that
Here, is the set . A linear function is also called a homomorphism.
When both and are linear spaces, then being a linear function is equvalent to saying that are collinear iff are collinear.
If is a linear space, then so is . This shows that if is onto, is a linear space if is.
Let be a one-to-one linear function. If points lie on line , then lie on . This also shows that three collinear points in are mapped to three collinear points in . In addition, we have
for any line in .
Definition. When is a bijection whose inverse is also linear, we say that is an isomorphism. When , we call an automorphism, or more commonly among geometers, a collineation, of the space .
Suppose is an isomorphism. For every point , let be the set of all lines passing through . Then
for any point in .
It is possible to have a bijective linear function whose inverse is not linear. For example, let be the space with two points with no lines, and the space with the same two points with line . Then the identity function on 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 |