# If $u,v,w$ are linearly independent, then is it true that $Tu, Tv, Tw$ are linearly independent?

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?

• Take $T=0$ as an example – Nick Feb 25 at 14:49

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