# Annihilator of image $T$ equals to $\text{ker}(T^*)$ [duplicate]

Let $T:V\to W$ be a linear transformation, then $\text{Ann}(\text{Im}T)=\text{ker}T^*$.

How one could start to prove it?

Many thanks.

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## marked as duplicate by Najib Idrissi, Davide Giraudo, 6005, Alex M., Claude LeiboviciFeb 1 at 11:04

$\phi\in Ann(ImT)$ iff $\phi(Tx)=0$ for all $x\in v$ iff $T^*(\phi) x=0$ for all $x\in V$ iff $\phi\in Ker(T^*)$