In the proof of the generalized Riemann extension theorem in Daniel Huybrechts's Complex Geometry, which is:
Proposition 1.1.7 (Riemann extension theorem) Let $f$ be a holomorphic function on an open set $U\subset \mathbb{C}^n$. If $g:U\backslash Z(f)\to\mathbb{C}$ is holomorphic and locally bounded near $Z(f)$ , then $g$ can uniquely be extended to a holomorphic function $\widetilde{g}:U\to\mathbb{C}.$
Where $Z(f)$ is defined to be $\{ z|f(z)=0 \}$.
Before starting the rigor proof, the author had assumed that:
...We can restrict to the case that the restriction $f_0$ of $f$ to this line (which means the set $ U\cap\{ (z_1,0,\dots,0) |z_1\in \mathbb{C} \} $) vanishes only in the origin. ...
Where $f_w$ is defined by $f_w(z_1)=f(z_1,w)$.
I guess this restriction could be made under a inversible linear transformation of $\mathbb{C}^n$ to itself. But is the transformation alway exists?
Could anyone give me some details about that? Why there is no loss of generality about this?
Yes, locally this always exists. This comes from the proof of Weierstrass Preparation Theorem.
You can always make a generic choice of a line that intersects $Z(f)$ only at $0$ up to restricting to a smaller open set $V\subset U$. After a linear change of coordinates you can assume that this line is the $z_1$-axis.