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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?

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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 $\cos^2\theta$?

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    @Qiaochu. Btw, yours is a nice example with the Vandermonde matrix.2010-10-20
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The cute application of the Wronskian I like to keep in mind is that the invertibility of the Vandermonde matrix implies that the set $\{ e^{\lambda z} : \lambda \in \mathbb{C} \}$ is linearly independent. The Wronskian also shows up, for example, in the method of variation of parameters.

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    No, they can be complex-valued real functions. Or you can pretend I said lambda in R. (This is less fun, though, because there's a fairly simple way to prove this directly: if you have any purported linear dependence, taking z large enough is a contradiction.)2010-10-20
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It can be concealed in various ways. If the functions are expressed as trig functions the dependence gets hidden easily. Think of $\sin^2(x)$ and $\cos(2x)$. As the functions get messier it gets easier to hide.

The problem with the Wronskian is that the functions must be sufficiently differentiable and you need to be able to calculate it. Think of the indicator function on the rationals and the indicator function on the irrationals. These are dependent, but the Wronskian won't help.

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    As with the trig example, you need a constant function for them to sum to zero. You can't compute the Wronskian here because you can't compute the derivatives.2016-10-03