A basic doubt on linear maps

A basic doubt on linear maps

Let $f$ be a linear map from $V$ to its field set, say $F$. Now if
$f(v)=0$ does it necessarily imply that $v=0$. I don't think so. $v=0$ is
one solution, but it need not be the only solution because $f$ need not be
one-to-one. Is this correct ?