
Journal of Convex Analysis 17 (2010), No. 1, 293299 Copyright Heldermann Verlag 2010 A Characterization of Injective Linear Transformations SoonMo Jung Mathematics Section, College of Science and Technology, HongIk University, 339701 Chochiwon, Republic of Korea smjung@hongik.ac.kr [Abstractpdf] 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. Keywords: Linear transformation, order relation, convexity. MSC: 15A04, 52A20 [ Fulltextpdf (102 KB)] for subscribers only. 