# transformation matrices and complex functions as projections

This question is about the connection between linear algebra and complex analysis. Coming from a two real dimensional domain a transformation matrix geometrically transforms a set of points (e.g. a circle) in that plane to another set of points in the two real dimensional codomain. The same holds true for a complex valued function which does the transformation from a one complex dimensional domain to the complex dimensional codomain.

I have two questions:
1. Do you know of any software (or add-ons) that let you "paint" some images and have them transformed by an arbitrary transformation matrix and/or complex function?
2. What is the connection between transformation matrices and complex functions? How can you get from one to the other, i.e. finding corresponding transformation matrices to a given complex function and vice versa to produce the same transformation on the plane?

Any answers concerning the above questions will be appreciated! Thank you!

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Geometer's Sketchpad 5 can probably apply almost any transformation (or complex function that can be used as a transformation) to an arbitrary image, including a hand-drawn one, but it's not necessarily a simple thing to achieve in Sketchpad. –  Isaac Jan 6 '11 at 15:39

2. In general, there is no connection. Transformation matrices correspond to linear mappings, and complex functions are in general nonlinear. Exception: the function $f(z)=(a+ib)z$ which simply multiplies $z$ by the constant $a+ib$ (where $a$ and $b$ are real) corresponds to the linear transformation given by the matrix $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$.
(Close to a point $z_0$ where $f'(z_0) \neq 0$, the formula $f(z)-f(z_0) \approx f'(z_0) (z-a)$ shows that the function $f$ can be approximated using a linear function of the above type, but that's a different story.)