If two functions are linearly dependent on an interval then prove that their wronskian is identically zero on that interval.

Question

If two functions $\phi_1(x) , \phi_2(x) $ are linearly dependent on an interval $I$ then prove that their wronskian $W(\phi_1,\phi_2)(x)=0 \forall x \in I$. The functions need not be solutions for an equation. Preferably without using eigenvectors and eigenvalues.

Answer 1

Linearly dependent means that there exist $\alpha,\beta$, not both zero, with $\alpha\phi_1+\beta\phi_2=0$. This implies also that $\alpha\phi_1'+\beta\phi_2'=0$. Then $$ \begin{bmatrix} \phi_1&\phi_2\\ \phi_1'&\phi_2'\end{bmatrix} \begin{bmatrix} \alpha\\ \beta\end{bmatrix} =\begin{bmatrix} 0\\0\end{bmatrix}. $$ So $\begin{bmatrix} \alpha\\ \beta\end{bmatrix}$ is an eigenvector of $W(\phi_1,\phi_2)$ for the eigenvalue $0$ (for any $x$). In particular, $$ \det W(\phi_1,\phi_2)=0. $$