If the images of vectors are linearly dependent, then they are linearly dependent

Question

I know that if the images of vectors are linearly independent, then the vectors are linearly independent. But will the statement still hold if we change independent to dependent? I tried testing with common linear transformations and so far the case holds. Are there any counter examples to it?

Answer 1

No. Take $f:\mathbb R\to\mathbb R$ given by $f(x)=0$. Then $\{f(1)\} = \{0\}$ is linearly dependent, but $\{1\}$ is linearly independent.