A function of the form f(z)=az+b where a and b are complex constants is called a linear transformation. show that every linear transformation can be expressed as the composition of a magnification, a ratation, and a translation. deduce from this that a linear transformation maps lines to lines and circles to circles.
Hint: write a in polar form.
If and then is magnification by . If then is minification by . If then is the same effect of however the object is reflected through the origin. And of course if then nothing happens.
Let then is a rotation counterclockwise by . If if then is a translation in direction of (if you think of this complex number as a vector).
Therefore, given where . So first you send then and then . Thus, all we are doing are rotating, maginifying/minifying, and translating. Thus, lines and circles go to lines and circles, respectively.