The Real And Complex Number Systems

W x
f 1 f 2 ... f n
... ... ... ...
f 1
n2
f 1
n
f 2
n2
f 2
n
... f n
n2
... f n
n
.
(b) Assuming the existence of n constants c 1 ,...,c n , not all zero, such that
c 1 f 1 x ...c n f n x 0 for every x in a, b, show that Wx 0 for each x in a, b.
Proof: Since c 1 f 1 x ...c n f n x 0 for every x in a, b, wherec 1 ,...,c n , not all
zero. Without loss of generality, we may assume c 1 0, we know that
c 1 f k 1 x ...c n f k n x 0 for every x in a, b, where 0 k n. Hence, we have
Wx
f 1 f 2 ... f n
f 1 f 2 ... f
n
... ... ... ...
f 1
n1
f 2
n1
... f n
n1
0
since the first column is a linear combination of other columns.
Note. A set of functions satisfing such a relation is said to be a linearly dependent set
on a, b.
(c) The vanishing of the Wronskian throughout a, b is necessary, but not sufficient.
for linear dependence of f 1 ,...,f n . Show that in the case of two functions, if the Wronskian
vanishes throughout a, b and if one of the functions does not vanish in a, b, then they
form a linearly dependent set in a, b.
Proof: Let f and g be continuous and differentiable on a, b. Suppose that fx 0for
all x a, b. Since the Wronskian of f and g is 0, for all x a, b, wehave
fg f g 0 for all x a, b. *
Since fx 0 for all x a, b, wehaveby(*),
fg f g
0 g
0 for all x a, b.
f 2 f
Hence, there is a constant c such that g cf for all x a, b. Hence, f, g forms a
linearly dependent set.
Remark: This exercise in (b) is a impotant theorem on O.D.E. We often write (b) in
other form as follows.
(Theorem) Letf 1 ,..,f n be continuous and differentiable on an interval I. If
Wf 1 ,...,f n t 0 0forsomet 0 I, then f 1 ,..,f n is linearly independent on I
Note: If f 1 ,..,f n is linearly independent on I, ItisNOT necessary that
Wf 1 ,...,f n t 0 0forsomet 0 I. For example, ft t 2 |t|, andgt t 3 . It is easy to
check f, g is linearly independent on 1, 1. AndWf, gt 0 for all t 1, 1.
Supplement on Chain Rule and Inverse Function Theorem.
The following theorem is called chain rule, it is well-known that let f be defined on an
open interval S, letg be defined on fS, and consider the composite function g f defined
on S by the equation