Geodesic
A Characterization of Injective Linear Transformations
Journal of convex analysis, Tome 17 (2010) no. 1, pp. 293-299

Voir la notice de l'article provenant de la source Heldermann Verlag

We prove a characterization of the injective linear transformations on real vector spaces: Let $X$ and $Y$ be an $m$-dimensional and an $n$-dimensional real vector spaces $(n \geq m \geq 2)$, respectively. Assume that a mapping $f \colon X \to Y$ satisfies ${\rm dim} f(X) \geq 2$ and $f(o) = o$, where $o$ denotes the origin of $X$ and $Y$. Then, $f$ is an injective linear transformation if and only if $f$ maps every line in $X$ onto a (corresponding) line in $Y$ and preserves the ordering on line.
Classification : 15A04, 52A20
Mots-clés : Linear transformation, order relation, convexity 
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     author = {S.-M. Jung},
     title = {A {Characterization} of {Injective} {Linear} {Transformations}},
     journal = {Journal of convex analysis},
     pages = {293--299},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {2010},
     url = {http://geodesic.mathdoc.fr/item/JCA_2010_17_1_JCA_2010_17_1_a20/}
}
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S.-M. Jung. A Characterization of Injective Linear Transformations. Journal of convex analysis, Tome 17 (2010) no. 1, pp. 293-299. http://geodesic.mathdoc.fr/item/JCA_2010_17_1_JCA_2010_17_1_a20/