Is linear map f * f = f injective? Let's say that $V$ is Vector Space and $f : V \to V$ is linear map with $f \circ f=f$  

I have to define what is $ker(f)$ of this function. 

Can we say that this function is also injective and so $ker(f) = {0} $?
Hint: $f$ injective and $f \circ f=f$ iff $f=\text{id}$.
If f•f=f and f is a linear transformation then there are two possibilities for f, either f is zero transformation or identity transformation.
If f is zero transformation then it is not injective and Ker(f)=V.
If f is an identity transformation then it is one one and ker(f)={0}.