True/False: If the Wronskian of n functions vanishes at all points on the real line then these functions must be linearly dependent in R.

Question

I know that if a set of functions are linearly dependent, then its Wronskian = 0 at all values of t in the interval.

So can you conclude that if Wronskian = 0 for all values of t in the interval, then the functions must be dependent?

Answer 1

The answer is no. For instance, the functions $f_1(x) = x^2$ and $f_2(x) = x \cdot |x|$ are continuous with continuous derivatives, have a Wronskian that vanishes everywhere, but fail to be linearly dependent.

The Wronskian Wikipedia page has a good discussion about this. Note that if the set of functions considered is analytic, then their dependence over an interval is indeed equivalent to their having a Wronskian that is identically zero.