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!