Can non-linear transformations be represented as Transformation Matrices? 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?
As Harry says, you can't (the example of affine transformations can be tweaked to work because they're just linear ones with the origin translated). However, approximating a nonlinear function by a linear one is something we do all the time in calculus through the derivative, and is what we often have to do to make a mathematical model of some real-world phenomenon tractable.
No.  Everything is determined by a choice of basis.  For a more in-depth answer, I would need to explain the first two weeks of linear algebra and draw some commutative diagrams.

If you'd like a better explanation, see pages 12-14 of Emil Artin's monograph Geometric Algebra.