Unitary coordinate transformation = rotation?

[Pacopag]

Homework Statement

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?

Homework Equations

The Attempt at a Solution

 

[Dick]

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

 

[Pacopag]

Thank you.