# If the images of vectors are linearly dependent, then they are linearly dependent [duplicate]

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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?

## marked as duplicate by Chinnapparaj R, Lord Shark the Unknown, kelvinn aja, José Carlos Santos linear-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Nov 22 '18 at 8:28

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• Try a projection. – Ramiro de la Vega Nov 21 '18 at 19:43

## 1 Answer

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.