You know that unitary transformations preserve length, don't you? Can't you use that?
Let be the linear transformation given by where . Demonstrate that T is unitary if and only if .
If is a unidimensional vector space, demonstrate that the only unitary linear transformations are the one given previously. In particular, if is a unidimensional vector space, there exist only 2 orthogonal transformations: and .
Attempt: Not much. I want to prove first.
If T is unitary, then . I don't know how to go further, I must show that this implies that .
But I'll try to follow your tip anyway. If I have any problem I'll repost.
Edit: Ok so if I understand well, A would be the matrix representative of T. We have ||Ax||=|c||x|=|x|, therefore T preserves lengths and so is unitary. Is this a valid proof?
Ok, so suppose matrix is unitary. That is, assume Let be a nonzero vector. Then
Therefore, since and we may conclude that for all nonzero . If the result is also true.
This proof you can use as an intermediate step in the proof you're ultimately interested in.
Does this help?
Yes it helped Ackbeet. If I'm not wrong, you're reasoning is for the part of the proof, right? What you've showed is equivalent to say , for all x. Which completes the proof for the part. I hope I'm not wrong on this.
For part , is my "proof" of post #5 valid?
Right. I think you've correctly finished the forward direction, if you include the lemma I proved in your proof.
The reverse direction I'm not so sure of. You have to show that, assuming there exists such that , such that for all vectors , that
So, assume the hypotheses with all the labels the same. I think you might also be able to assume that exists. You must show that
Can you compute and directly? I think you might be able to do that. Then you simply show that they are the same.