Why is L(z)=Az+B a linear transformation in Complex Analysis?

Question

Recently, I am learning complex analysis using " Complex Analysis for mathematics and engineering" by John H. Mathews and Russell W. Howell. In Chapter 2, it says that "...because a linear transformation can be considered to be a composition of a rotation, a magnification, and a translation", where in the text a linear transformation is defined as w=L(z)=Az+B. This is really confusing, because in linear algebra, this doesn't make sense. So, is the definition of linear transformation in complex number different from that of linear algebra? Thank you!

Answer 1

There is a very sad confusion that arises naturally as a result of poor choice of terminology in English mathematics, which is calling functions of the form $ f (x)=ax+b $ linear functions - $ x $ be a real or complex number, it doesn't matter.

The reason for this is that those are functions whose graph is a line (be it a real or complex line).

The reason it is unfortunate is that these functions are not necessarily linear transformations, since $ f (0)=b $ is not necessarily $0 $.

But note that they are however affine transformations, which means that if they don't necessarily respect linear combinations, they do respect affine combinations. Other languages would more appropriately call such functions affine functions.