A Characterization of Injective Linear Transformations

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.