[Math] Prove: $ \operatorname{Ker}(T)^\perp= \operatorname{Im}(T^*)$

Question

Let $T:V\to V$

Prove: $ \operatorname{Ker}(T)^\perp= \operatorname{Im}(T^*)$

If $v\in \operatorname{Im}(T^*)$ so $\exists w\in V:T^*w=v$ but how can I continue from here?

If $v\in \operatorname{Ker}(T)^\perp$ what does it say?

Answer 1

We have

\begin{align} x \in \ker T &\iff Tx = 0 \\ &\iff \langle Tx,y\rangle= 0, \forall y \in V \\ &\iff \langle x,T^*y\rangle= 0, \forall y \in V \\ &\iff x \perp \operatorname{Im} T^* \\ &\iff x \in (\operatorname{Im} T^*)^\perp \end{align}

so $\ker T = (\operatorname{Im} T^*)^\perp$. Now take the orthogonal complement.