20
$\begingroup$

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?

  • 5
    With what definitions of "rotation" and "linear transformation" are you working?2010-08-24
  • 0
    A sketch: matrices represent linear transformations, and rotations can be represented by matrices. A bit handwave-y though.2010-08-24
  • 0
    @J. Mangaldan: this is just restating the question.2010-08-25
  • 0
    As I said... handwave-y. Looking at it again, circular too. :)2010-08-25
  • 0
    To help ensure non-circularity of the definitions, I'd start of with the theorem: an isometry of the plane (as a metric space) is either fixed-point free, has a fixed line or has a single fixed point. Defn: a "rotation" about a point is an isometry of the plane that has only that point as its fixed-point set.2010-08-25
  • 1
    With these definition's *guest*'s question becomes: if $f:\mathbb R^2 \to \mathbb R^2$ is an isometry of metric spaces such that $f(0)=0$, $f(1)=1$ and $f(i)=i$, then $f=Id_{\mathbb R^2}$. And proving reduces quickly to the issue that straight lines are the unique continuous curve that minimizes length between points. A standard (and non-trivial) thing to prove.2010-08-25

7 Answers 7