Page 125 - A Course in Linear Algebra with Applications

4.3: Linear Independence in Vector Spaces 109

linear system has only the trivial solution and the functions
/i) /2 5 • • •, fn w m De linearly independent. Define

h h In

W(f 1,f 2,...,f n) = fi f
J n
, ( n - l ) ( n - l ) / ( n - l )
/: Jn

This determinant is called the Wronskian of the functions
/i ? /2, • • •, fn • Then our discussion shows that the following is
true.

Theorem 4.3.2 Suppose that fi, f 2, • • •, f n are functions
whose first n — 1 derivatives exist in the interval [a,b\. If
W(fi, / 2 , . . . , f n) is not identically equal to zero in this inter-
val, then i , f 2 , . . . , f n are linearly independent in [a, b}.
/
The converse of 4.3.2 is false. In general one cannot
conclude that if i , f 2, • • •, f n are linearly independent, then
/
W(fi,f2, • • •, f n) is not the zero function. However, it turns
out that if the functions fi,f 2,---,fn a r e solutions of a ho-
mogeneous linear differential equation of order n, then the
Wronskian can never vanish. Hence a necessary and sufficient
condition for a set of solutions of a homogeneous linear differ-
ential equation to be linearly independent is that their Wron-
skian should not be the zero function. For a detailed account
of this topic the reader should consult a book on differential
equations such as [16].

Example 4.3.3
x 2x
Show that the functions x,e ,e~ are linearly indepen-
dent in the vector space C[0, 1].
The Wronskian is

e-2x
lx
x
W(x,e , e~ ) = -2e- 2x 3(2x-l)e"
4 e - 2 *