Why is the Mobius transform called a linear transform?

Question

It has the form

$$ z \rightarrow w = \frac{a z +b}{cz +d } . $$

This is apparently not a linear function. A linear function should be in the form of

$$ w = a z +b ,$$

right?

ps: I am reading the book by Ahlfors.

Answer 1

You're right: Ahlfors calls it a linear transformation.

The current term is now linear fractional transformation.