Does this theorem need that Ker{F}=0?

[jamalkoiyess]

I have encountered this theorem in Serge Lang's linear algebra:
Theorem 3.1. Let F: V --> W be a linear map whose kernel is {O}, then If v1 , ... ,vn are linearly independent elements of V, then F(v1), ... ,F(vn) are linearly independent elements of W.

In the proof he starts with C1F(v1) + C2F(v2) + ... + CnF(vn) and then uses linearity and injectivity to prove that the constants are 0.

I can't see where injectivity is essential, he could have proved it with linearity alone.
He arrives at a point where we have :

F(C1V1 + ... + CnVn) = 0

He uses injectivity here to prove that C1V1 + ... + CnVn = 0 and since v1 ... vn linearly independent, the constants are 0. But that can be also solved by the fact that F is linear.

 

[martinbn]

Say $F(v_1)=0$, then $1*F(v_1)+0*F(v_2)+\cdots+0*F(v_n)=0$, they are dependent.