Now I'm confused with what "a linear transformation" means.
In linear algebra textbook, I learned that a linear transformation is $T:V \to W$, where V,W are vector spaces, which satisfies additivity and homogeneity, in other words, $T(u+v)=Tu+Tv, T(av)=aTv$ for all $u,v \in V$ and $a \in F$
But in my complex analysis textbook, $\displaystyle f(z)=\frac{az+b}{cz+d}$, $a,b,c,d \in \mathbb C$, is introduced as an example of a linear transformation.
However, this function $f$ doesn't seem to follow the definition from linear algebra. Indeed, $f(0) \neq 0$.
Is it like there are two kinds of linear transformations in mathematics, or they are actually the same thing but I don't get it well?