I am trying to prove a theorem, it states the following:
"Let $V, W$ be vector spaces and $T: V \rightarrow W$ a linear transformation, then $T$ is injective iff $T$ preserves linear independence".
I already proved the first implication, but I'm stuck on the latter. I'm trying to use the fact that a transformation is injective iff $\ker(T) = \{0\}$ to prove injectivity.
I also tried with the idea of proposing a base (by definition linearly independent) and write a vector in the kernel as a combination of the elements of the base. I was told this implicitly assumes finite dimensionality which is not included in the hypothesis.
Am I on the right track? Or do I need stronger theorems to prove this statement given these hypothesis?
Best Answer
If $T$ is not injective, there is a non-null vector $v$ such that $T(v)=0$. So, $\{v\}$ is linearly independent, but $T\bigl(\{v\}\bigr)$ is not.