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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?

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marked as duplicate by Chinnapparaj R, Lord Shark the Unknown, kelvinn aja, José Carlos Santos linear-algebra Nov 22 '18 at 8:28

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    $\begingroup$ Try a projection. $\endgroup$ – Ramiro de la Vega Nov 21 '18 at 19:43
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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.

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