The
functions
,
, ...,
are linearly dependent if, for some
,
, ...,
not all zero,
![\begin{displaymath}
c_if_i(x) = 0
\end{displaymath}](l2_344.gif) |
(1) |
(where Einstein Summation is used) for all
in some interval
. If the functions are not linearly dependent,
they are said to be linearly independent. Now, if the functions
, we can differentiate (1) up
to
times. Therefore, linear dependence also requires
![\begin{displaymath}
c_i f_i' = 0
\end{displaymath}](l2_348.gif) |
(2) |
![\begin{displaymath}
c_i f_i'' = 0
\end{displaymath}](l2_349.gif) |
(3) |
![\begin{displaymath}
c_i f_i^{(n-1)} = 0,
\end{displaymath}](l2_350.gif) |
(4) |
where the sums are over
, ...,
. These equations have a nontrivial solution Iff the Determinant
![\begin{displaymath}
\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}](l2_351.gif) |
(5) |
where the Determinant is conventionally called the Wronskian and is denoted
. If
the Wronskian
for any value
in the interval
, then the only solution possible for (2) is
(
, ...,
), and the functions are linearly independent. If, on the other hand,
for a range, the
functions are linearly dependent in the range. This is equivalent to stating that if the vectors
,
...,
defined by
![\begin{displaymath}
{\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}](l2_358.gif) |
(6) |
are linearly independent for at least one
, then the functions
are linearly independent in
.
References
Sansone, G. ``Linearly Independent Functions.'' §1.2 in Orthogonal Functions, rev. English ed.
New York: Dover, pp. 2-3, 1991.
© 1996-9 Eric W. Weisstein
1999-05-25