Hi,
exist a special case, where we have a map which is completly linear that is not injective?
greetings
p.s.
I heard that injective in english is also translated to "one-to-one" Is this correct?
The zero map is a linear map, and it is not injective!
In the particular case of linear maps from $\displaystyle \mathbb{R}$ to $\displaystyle \mathbb{R}$, a linear map is injective (and even bijective) iff it is a non constant map.
In a more general case, linear map between two vector spaces can be non injective without being a constant map, for exemple
$\displaystyle f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}x,y)\mapsto (x+y,0)$
p.s. Yes, one-to-one means injective.