The wronskian determinant of $n$ functions which are $(n-1)$ times differentiable is
$$W(f_1,\dotsc,f_n)=\det\begin{pmatrix}
f_1 & f_2 & \dots & f_n\\
f_1' & f_2' & \dots & f_n'\\
\vdots & & & \vdots\\
f_1^{(n-1)} & f_2^{(n-1)} & \dots & f_n^{(n-1)}
\end{pmatrix}.$$

It is clear that if $f_1$, ..., $f_n$ are linearly dependent, $W(f_1,\dotsc,f_n)=0$. For some years, the converse was assumed to be true too, until Peano gave the counterexample: $f_1=x^2$ and $f_2=x\cdot |x|$ are linearly independent, though $W(f_1,f_2)=0$ everywhere. Later, Bôcher even gave counter examples with infinitely differentiable functions.

Bôcher also proved that the converse holds as soon as the functions are analytic. Other conditions are also known for the converse to hold.

Engdahl and Parker describe the history of the wronskian [1]. For a nice proof of Bôcher's result, one can have a look at a paper of Bostan and Dumas [2].

[1] Susannah M. Engdahl and Adam E. Parker. **Peano on Wronskians: A Translation**, *Loci* (April 2011), DOI:10.4169/loci003642.

[2] Alin Bostan and Philippe Dumas. **Wronskians and linear independence**, *American Mathematical Monthly*, vol. 117, no. 8, pp. 722–727, 2010.