TITLE: Example of functions where linear dependence isn't obvious
QUESTION [5 upvotes]: The Wronskian lets us determine if a set of functions (possibly the solutions to a differential equation) are linearly dependent or not. But, for every example in the book, it is very obvious if one of the functions is a linear combination of the others. The examples in the book use 3-5 functions. What would be an example of a small number of functions where this isn't obvious?
Or is the application of the Wronskian mostly to deal with large sets of functions... where the sheer number makes it hard to tell if they are dependent or not?

REPLY [3 votes]: Have you tried to prove by hand (i.e., only using the definition of linear independence)  that $\sin \theta$ and $ \cos \theta$ are linearly independent? Of course, this can be done with the help of the Wronskian.
And what about $e^{i\theta}$ and $ e^{i (\theta + \frac{\pi}{2})}$? This is geometrically clear, but: can you see the difference between linear independence over the real and complex numbers?
EDIT. Just to add still a more elementary example: what about $\sin^2\theta $ and $\cos^2\theta$?