linear function

Let ๐’ฎ1=(๐’ซ1,โ„’1) and ๐’ฎ2=(๐’ซ2,โ„’2) be two near-linear spaces.

Definition. A linear functionMathworldPlanetmath from ๐’ฎ1 to ๐’ฎ2 is a mapping on the points that sends lines of ๐’ฎ1 to lines of ๐’ฎ2. In other words, a linear function is a function ฯƒ:๐’ซ1โ†’๐’ซ2 such that

ฯƒโข(โ„“)โˆˆโ„’2โขย for everyย โขโ„“โˆˆโ„’1.

Here, ฯƒโข(โ„“) is the set {ฯƒโข(P)โˆฃPโˆˆโ„“}. A linear function is also called a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

When both ๐’ฎ1 and ๐’ฎ2 are linear spacesPlanetmathPlanetmath, then ฯƒ being a linear function is equvalent to saying that P,Q are collinearMathworldPlanetmath iff ฯƒโข(P),ฯƒโข(Q) are collinear.

If ๐’ฎ1 is a linear space, then so is (ฯƒโข(๐’ซ1),ฯƒโข(โ„’1)). This shows that if ฯƒ:๐’ฎ1โ†’๐’ฎ1 is onto, ๐’ฎ2 is a linear space if ๐’ฎ1 is.

Let ฯƒ:๐’ฎ1โ†’๐’ฎ2 be a one-to-one linear function. If points P1โ‰ P2 lie on line โ„“, then ฯƒโข(P1)โ‰ ฯƒโข(P2) lie on ฯƒโข(โ„“). This also shows that three collinear points in ๐’ฎ1 are mapped to three collinear points in ๐’ฎ2. In addition, we have

|โ„“|=|ฯƒโข(โ„“)| for any line โ„“ in ๐’ฎ1.

Definition. When ฯƒ:๐’ฎ1โ†’๐’ฎ2 is a bijectionMathworldPlanetmath whose inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ฯƒ-1 is also linear, we say that ฯƒ is an isomorphismPlanetmathPlanetmath. When ๐’ฎ1=๐’ฎ2=๐’ฎ, we call ฯƒ an automorphism, or more commonly among geometers, a collineationMathworldPlanetmath, of the space ๐’ฎ.

Suppose ฯƒ:๐’ฎ1โ†’๐’ฎ2 is an isomorphism. For every point P, let P* be the set of all lines passing through P. Then

|P*|=|ฯƒโข(P)*| for any point P in ๐’ฎ1.

It is possible to have a bijective linear function whose inverse is not linear. For example, let ๐’ฎ1 be the space with two points P,Q with no lines, and ๐’ฎ2 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 โ€œlinear functionโ€ differs from its more usual meaning as a linear transformation between vector spaces in the study of linear algebra.


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