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 $f\_1(x),f\_2(x),…,f\_n(x), x\in I$:

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> 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 $I$:
>  $$ \det\left( \begin{array}{ccccc} \int\_{a}^{b} f\_1^2 dx& \int\_{a}^{b} f\_1f\_2 dx&… &\int\_{a}^{b}f\_1f\_ndx \\ \int\_{a}^{b}f\_2f\_1dx & \int\_{a}^{b}f\_2^2 dx &... &\int\_{a}^{b}f\_2f\_ndx \\ ⋮ & ⋮ & ⋮ &⋮ \\ \int\_{a}^{b}f\_nf\_1dx & \int\_{a}^{b}f\_nf\_2dx&... &\int\_{a}^{b}f\_n^2dx \end{array} \right) $$
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It seems to be a great practical Theorem, but I couldn't find its proof. I really appreciate your help.