Let $V$ and $W$ be two vector spaces, and let $f:V \rightarrow W$ be a linear mapping. Show that $f$ is a Monomorphism only when $f$ maps linearly independent sets in $V$ to linearly independent sets in $W$.
I sort of understand the idea as to why in the above case $f$ should be a Monomorphism. According to my Linear Algebra script, it describes a Monomorphism as an injective linear mapping. So, if my understanding is correct, if the vectors were not linearly independent in $W$ for example, then one vector in $V$ could map to several linearly dependent vectors in $W$ (which is not injective).
Is this an example of a proof where I just need to write out my reasoning, or is there a way to show that the above is true more "mathematically"?