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Linearly Dependent Functions

The $n$ functions $f_1(x)$, $f_2(x)$, ..., $f_n(x)$ are linearly dependent if, for some $c_1$, $c_2$, ..., $c_n\in
\Bbb{R}$ not all zero,

c_if_i(x) = 0
\end{displaymath} (1)

(where Einstein Summation is used) for all $x$ in some interval $I$. If the functions are not linearly dependent, they are said to be linearly independent. Now, if the functions $\in \Bbb{R}^{n-1}$, we can differentiate (1) up to $n-1$ times. Therefore, linear dependence also requires
c_i f_i' = 0
\end{displaymath} (2)

c_i f_i'' = 0
\end{displaymath} (3)

c_i f_i^{(n-1)} = 0,
\end{displaymath} (4)

where the sums are over $i=1$, ..., $n$. These equations have a nontrivial solution Iff the Determinant
\left\vert\matrix{f_1 & f_2 & \cdots & f_n\cr
f_1' & f_2' &...
...n-1)} & f_2^{(n-1)} & \cdots & f_n^{(n-1)}\cr}\right\vert = 0,
\end{displaymath} (5)

where the Determinant is conventionally called the Wronskian and is denoted $W(f_1,f_2,\ldots,f_n)$. If the Wronskian $\not = 0$ for any value $c$ in the interval $I$, then the only solution possible for (2) is $c_i =
0$ ($i=1$, ..., $n$), and the functions are linearly independent. If, on the other hand, $W = 0$ for a range, the functions are linearly dependent in the range. This is equivalent to stating that if the vectors ${\bf V}[f_1(c)]$, ..., ${\bf V}[f_n(c)]$ defined by
{\bf V}[f_i(x)] = \left[{\matrix{f_i(x)\cr f_i'(x)\cr f_i''(x)\cr \vdots\cr f_i^{(n-1)}(x)\cr}}\right]
\end{displaymath} (6)

are linearly independent for at least one $c \in I$, then the functions $f_i$ are linearly independent in $I$.


Sansone, G. ``Linearly Independent Functions.'' §1.2 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 2-3, 1991.

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