Hi.
What do you mean by completely linear?
The zero map is a linear map, and it is not injective!
In the particular case of linear maps from to , 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
x,y)\mapsto (x+y,0)" alt="f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}x,y)\mapsto (x+y,0)" />
p.s. Yes, one-to-one means injective.