TITLE: How do you formally prove that rotation is a linear transformation?
QUESTION [29 upvotes]: The fact that rotation about an angle is a linear transformation is both important (for example, this is used to prove the sine/cosine angle addition formulas; see How can I understand and prove the "sum and difference formulas" in trigonometry?) and somewhat intuitive geometrically.  However, even if this fact seems fairly obvious (at least from a diagram), how does one turn the picture proof into a formal proof?  On a related note, it seems likely that many formal proofs using a diagram will end up relying on Euclidean geometry (using angle/side congruence properties), but isn't one of the points of linear algebra to avoid using Euclidean geometry explicitly?

REPLY [3 votes]: As an exercise in trigonometry.

The transformation of rectangular co-ordinates $(x,y)$ of $P$ into the
rectangular co-ordinates $(x',y')$ of $P$ by a rotation of the axes is given by:
$x=x^{\prime }\cos \phi -y^{\prime }\sin \phi $
$y=x^{\prime }\sin \phi +y^{\prime }\cos \phi $.
This transformation is invertible.
Edit: This way of proving the transformation of rectangular co-ordinates does not "avoid using Euclidean geometry explicitly" (in this case trigonometry), as writen in the question.
However the same method can be used to derive the transformation of rectangular into spherical co-ordinates, which is a non-linear transformation.