# Unitary coordinate transformation = rotation?

1. Jun 8, 2008

### Pacopag

1. The problem statement, all variables and given/known data
Suppose I define a linear coordinate transformation that I can describe with a matrix U.
If U is unitary. i.e.
$$U^{-1}U = UU^{-1}=1$$
does that necessarily imply that the transformation corresponds to a pure rotation (plus maybe a translation), so that I may assume that volumes are invariant?

2. Relevant equations

3. The attempt at a solution

2. Jun 8, 2008

### Dick

Yes, volumes are invariant, certainly. The jacobian is 1. There are no translations if the transformation is linear. There could be reflections.

3. Jun 9, 2008

Thank you.