A Characterization of Injective Linear Transformations

We prove a characterization of the injective linear transformations on real vector spaces: Let

and

Y Y

be an

m m

-dimensional and an

n n

-dimensional real vector spaces

(n \geq m \geq 2) ( n ≥ m ≥ 2 )

, respectively. Assume that a mapping

f \colon X \to Y f  ⁣ : X → Y

satisfies

{\rm dim} f(X) \geq 2 d i m f ( X ) ≥ 2

and

f(o) = o f ( o ) = o

, where

o o

denotes the origin of

X X

and

Y Y

. Then,

f f

is an injective linear transformation if and only if

f f

maps every line in

X X

onto a (corresponding) line in

Y Y

and preserves the ordering on line.