Linear Mapping Monorphism

Question

The question I have reads as follows:

Prove the following statement:

A linear map $f: V \to W$ is a monomorphism if and only if f(S) is Linearly Independent in W whenever S is Linearly Indepdendent in V.

I have tried working this out with linearity and through arguing that in a monomorphism only the zero vector can map to the kernel of f but I still think I'm missing something.

Any help would be great.

Answer 1

For the one direction, consider $v \neq 0$, hence $\{v\}$ is independent. Now look at $\{f(v)\}$.

For the other direction, let $\{v_i\}_i$ be independent. Consider a linear combination $\sum a_if(v_i)$, use linearity, and then deduce $\sum a_iv_i =0$, hence all $a_i=0$.