A Characterization of Injective Linear Transformations

Abstract

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.

Contact details are reproduced from the original publication and may be historical.

Mathematics Section, College of Science and Technology, Hongik University, 339-701 Jochiwon, Republic of Korea

15A04, 52A20

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.