Unitary transformation on vector is equal to the transpose on its orthogonal vector?

Question

While working on my Quantum course, I have observed that

U|0> = U^T|1>

for a Unitary matrix U. This is solely based on observations and calculating the matrices. I wish to try and prove this mathematically but haven't got any conclusive results.

Things I tried - using general properties of unitary matrices, I found that

U^T = ((complex conjugate)U)^-1

But can't seem to prove my observation.

My question here is

1. Is what I concluded correct for the orthogonal vectors of |0> and |1>?
2. Whether this property can be extended to all orthogonal vectors?
3. How can I prove it mathematically?

Answer 1

Your observation seems to be incorrect: if $U$ is the identity operator (which is certainly unitary), then $U|0\rangle=U^T|1\rangle$ reduces to $|0\rangle=|1\rangle$, which is false.