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,

$$c_if_i(x) = 0$$ (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

$$c_i f_i' = 0$$ (2)

$$c_i f_i'' = 0$$ (3)

$$c_i f_i^{(n-1)} = 0,$$ (4)

where the sums are over $i=1$, ..., $n$. These equations have a nontrivial solution Iff the Determinant

$$\left\vert\matrix{f_1 & f_2 & \cdots & f_n\cr f_1' & f_2' &... ...n-1)} & f_2^{(n-1)} & \cdots & f_n^{(n-1)}\cr}\right\vert = 0,$$ (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

$${\bf V}[f_i(x)] = \left[{\matrix{f_i(x)\cr f_i'(x)\cr f_i''(x)\cr \vdots\cr f_i^{(n-1)}(x)\cr}}\right]$$ (6)

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

References

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