I just came back from an intense linear algebra lecture which showed that linear transformations could be represented by transformation matrices; with more generalization, it was later shown that affine transformations (linear + translation) could be represented by matrix multiplication as well.
This got me to thinking about all those other transformations I've picked up over the past years I've been studying mathematics. For example, polar transformations -- transforming $x$ and $y$ to two new variables $r$ and $\theta$.
If you mapped $r$ to the $r$ axis and $\theta$ to the $y$ axis, you'd basically have a coordinate transformation. A rather warped one, at that.
Is there a way to represent this using a transformation matrix? I've tried fiddling around with the numbers but everything I've tried to work with has fallen apart quite embarrassingly.
More importantly, is there a way to, given a specific non-linear transformation, construct a transformation matrix from it?