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Journal of Convex Analysis 17 (2010), No. 1, 293--299
Copyright Heldermann Verlag 2010

A Characterization of Injective Linear Transformations

Soon-Mo Jung
Mathematics Section, College of Science and Technology, Hong-Ik University, 339-701 Chochiwon, Republic of Korea


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

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