[Math] If $u,v,w$ are linearly independent, then is it true that $Tu, Tv, Tw$ are linearly independent

Question

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?

Answer 1

Not necessary as you can see from Nick example.

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