Wronskian of Linearly Dependent Mappings is Zero

Theorem

Let $\map {f_1} x, \map {f_2} x, \dotsc, \map {f_n} x$ be real functions defined on a closed interval $\closedint a b$.

Let $f_1, f_2, \ldots, f_n$ be $n - 1$ times differentiable on $\closedint a b$.

Then:

$\map W x = 0$

if and only if:

there exists a subinterval of $\closedint a b$ on which $f_1, f_2, \ldots, f_n$ are linearly dependent.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Wronskian
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Wronskian