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Let $u,v$ and $w$ are vectors in a vector space $V$ and $T: V → W$ is a linear transformation. If $u,v,w$ are linearly independent, then is it true that $Tu, Tv, Tw$ are linearly independent?

Let $c_1,c_2,c_3$ be scalars such that $$c_1Tu+c_2Tv+c_3Tw = 0$$ Then $$T(c_1u+c_2v+c_3w) = 0$$ and from this we can conclude that $c_1u+c_2v+c_3w = 0$ and hence $c_1,c_2,c_3=0$ only when $T$ is invertible.

So in general $Tu, Tv, Tw$ are not linearly independent.

Is the solution correct?

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    $\begingroup$ Take $T=0$ as an example $\endgroup$ – Nick Feb 25 at 14:49
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No, this will fail for any non-injective $T$, try and think of a function that may send two linearly independent vectors to the same vector.

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Not necessary as you can see from Nick example.

This would be true only if linear transformation $T$ is injective, that is iff $\operatorname{Ker} T = \{0\}$.

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