Showing that $\left(\operatorname{Im}\left(T^*\right)\right)^\perp=(\ker T)$ (solution verification)

Question

I have been told that my proof is not true at all without any explanation why, I would like if any one can help.

Let $T: V\to V$ linear operator in finite inner product space, show that $$\left(\operatorname{Im}\left(T^*\right)\right)^\perp=(\ker T)$$

My attempt:

$\left(\operatorname{Im}\left(T^*\right)\right)^\perp=(\ker T) \iff \operatorname{Im}(T^*)^=\ker\left(T^{}\right)^\perp$

Let $\vec v\in Im(T^*) $ and let $\vec w \in \ker(T)$ so we can write :

$\langle T^*(v),w\rangle=\langle v,T(w)\rangle=\langle v,0\rangle=0$