Prove or dispove

$\displaystyle T:R^n \rightarrow R^n$ is a linear transformation

if for every $\displaystyle u \in R^n $ and for every $\displaystyle v \in KerT$,

$\displaystyle T(u) \cdot v=0$

then $\displaystyle KerT = (ImT)^{\perp}$

Attempt:

even though these questions are always false, sadly I chose True and gave an explanation why.

If this is false can someone please explain why and give an example.

(even if it's true can you explain why)

Thanks!