I am reading the paper "A refined version of the Siegel-Shidlovskii theorem" by F. Beukers. In the proof of Theorem 1.5, he mentions the following results in Galois differential theory. Let me formulate it as follows.
Consider a linear differential equation over a differential field $k$ of characteristic zero whose constant field $C$ is algebraically closed. Consider a linear differential equation in $Y$ $$Y'=AY(\star)\text{ where }A\in M_{n\times n}(k).$$ Let $G$ be its Galois differential group, so $G=\mathrm{Gal}_{\partial}(L/k)$ where $L$ is the Picard-Vessiot field. Let $\mathbf{Y}=(y_1,...,y_n)\in L^n$ be a solution of the equation $(\star)$ such that $y_1,\dots,y_n$ are linear independent over $k$. Then the images of $\mathbf{Y}$ under $G$ span the complete solution set of $(\star)$.
Here, is it true that we span the vector space over $C$? Can anyone provide me with some hints or any reference to this result? I think, in some sense, it is quite similar to Galois theory where the Galois group acts transitively on the set of solutions of the minimal equation.