Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $ \det\left( \begin{array}{ccccc} f_1 & f_2 & f_3 &… &f_n \\ f'_1 & f'_2 & f'_3 &... &f'_n \\ ⋮ & ⋮ & ⋮ &⋮ &⋮ \\ f_1^{(n-1)} & f_2^{(n-1)} & f_3^{(n-1)} &... &f_n^{(n-1)} \end{array} \right) $ called Wronskian of $f_1,f_2,…,f_n$ ,is not zero for at least one point in the interval $I$. Equivalently, if functions $f_1,f_2,…,f_n$ possess at least $n-1$ derivatives and are linearly dependent on $I$ then $W(f_1,f_2,…,f_n)(x)=0$ for every $x\in I$. So this equivalent statement gives just a necessary condition for dependency of above functions on the interval. Fortunately, there is necessary and sufficient condition for dependency of a set of functions $f_1(x),f_2(x),…,f_n(x), x\in I$:
A set of functions $f_1(x),f_2(x),…,f_n(x), x\in I$ is linearly dependent on $I$ iff the determinant below is identically zero on $I$: $ \det\left( \begin{array}{ccccc} \int_{a}^{b} f_1^2 dx& \int_{a}^{b} f_1f_2 dx&… &\int_{a}^{b}f_1f_ndx \\ \int_{a}^{b}f_2f_1dx & \int_{a}^{b}f_2^2 dx &... &\int_{a}^{b}f_2f_ndx \\ ⋮ & ⋮ & ⋮ &⋮ \\ \int_{a}^{b}f_nf_1dx & \int_{a}^{b}f_nf_2dx&... &\int_{a}^{b}f_n^2dx \end{array} \right) $
It seems to be a great practical Theorem, but I couldn't find its proof. I really appreciate your help.