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This question already has an answer here:

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 Leibovici Feb 1 at 11:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

up vote 4 down vote accepted

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

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