How the Wronskian works

Question

To prove linear independence of a set of functions, we say that given their Wronskian matrix $~W~$, $~W_x = 0~$ implies trivial solution $~(0,0,0,\cdots)~$ if the value (determinant) of the Wronskian is identically non-zero.

But why so?

For all I know, $~W = 0~$ only implies a unique solution, not necessarily $~(0,0,0,\cdots)~$.

All the articles on the web are really confusing me.

Answer 1

OK, so you know that there is only one unique solution to the system. Can you try guessing it? It's pretty trivial...