Is $T: V \rightarrow V$ always injective?
My thoughts would be that it is since $V$ is isomorphic to $V$.
There is a bijection. But I feel like I have seen examples where this is not true.
Am I missing something?
2026-03-27 19:31:37.1774639897
Is $T\in \mathcal{L}(V)$ always injective?
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As stated in the comments, $T$ does not need to be injective. However, assuming $\operatorname{dim}V < \infty$, it follows from the nullity rank theorem that $T: V \to V$ is injective iff $T$ is surjective.