A linear map sends a dependent list to a dependent list: apply it to a nontrivial relation and retain the same nonzero coefficients. For image sets, deduplicating equal images can invalidate the statement. Example: $e_1,e_2,2e_1-e_2$ are dependent, while $T(x,y)=x+y$ maps their set to $\{1\}$, which is independent in $\mathbb R$.
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