Definition
Let $\phi_1,\ldots,\phi_n\in C^n(I)$, and set
$$W(\phi_1,\ldots,\phi_n)(t)=\big(\phi_i^{(j-1)}(t)\big).$$
We define the Wronskian of $\phi_1,\ldots,\phi_n$ as $w(t)=\det W(t)$.
$\blacksquare$
Remarks
Fact I. If $w(t)$ is not identically zero in $I$ then $\phi_1,\ldots,\phi_n$ are linearly independent in $I$.
Fact II. The functions $t^2$ and $|t|\cdot t$ are linearly independent but their Wronskian is 0 everywhere. So non-null Wronskian implies independence but null Wronskian DO NOT implies linear independence.
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Author of the notes: Antonio J. Pan-Collantes
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