I am not sure how "transpose" is defined for non-Euclidean linear transformations. But if V is an Euclidean space then can be tought as where is a square matrix. Since this endomorphism has no trivial null space, i.e. has non-trivial solutions it means . But since it means has non-trivial solutions. And so has the property mentioned.
Suppose is non-trivial: there exists some such that . We are to find a nonzero functional such that ; Let be the length of the projection on (well defined). Then for all , we have or identically. So and is non-trivial.
For the converse, interchange between and .