I know that given two analytic functions on some domain $D$ of the complex plane, then their Wronskian determinant being $0$ is equivalent to them being linearly dependent. I would like to generalize this to a finite family of analytic functions. Naturally, one should use induction. However, the proof of $n =2$ is horrifically computational (for me, that is). Is there any clever way to avoid the mess in the induction process? Here are some notes that contain the proof of $n =2$ in case one would like to look at it.
Thank you very much in advance. I am much obliged.
Let $M$ be the matrix $\pmatrix{f_1 & \ldots & f_n\cr \ldots & \ldots & \ldots\cr f_1^{(n-1)} & \ldots & f_n^{(n-1)}\cr}$ whose determinant is the Wronskian.
Hint: the existence of a nonzero vector in the null space of the square matrix $M$ is equivalent to $\det M = 0$.