Suppose $V$ is a finite dimensional vector space over $\mathbb{K}$. Let $A$ be a linear map. I am trying to prove that $$\operatorname{Im}A^\text{T}=(\operatorname{Ker}A)^{\perp}$$
I know one direction: $\operatorname{Im}A^\text{T} \subset (\operatorname{Ker}A)^{\perp}$ but I don’t know how to show the other direction. Can anyone help me?