Page 124 - A Course in Linear Algebra with Applications

108 Chapter Four: Introduction to Vector Spaces

In 5.1 we shall learn how to tell if a set of vectors in an
arbitrary finitely generated vector space is linearly dependent.

An application to differential equations
In the theory of linear differential equations it is an im-
portant problem to decide if a given set of functions in the
vector space C[a, b] is linearly dependent. These functions
will normally be solutions of a homogeneous linear differential
equation. There is a useful way to test such a set of func-
tions for linear independence using a determinant called the
Wronskian.
Suppose that i , /2, • • •, /«, are functions whose first n—\
/
derivatives exist at all points of the interval [a, b\. In particular
this means that the functions will be continuous throughout
the interval, so they belong to C[a, b]. Assume that ci, C2,...,
are real numbers such that Ci/i +C2/2 + • • = 0, the
c n • + c nf n
zero function on [a ,b]. Now differentiate this equation n — 1
times, keeping in mind that the Cj are constants. This results
in a set of n equations for c\, c^,..., c n

= 0
{ c i / i + C2/2 + • • • + c nf n = 0
+
c nf n
+
cif{
+•••
c 2ti
n 1)
( 1) ( n 1 } + c nf n - = 0
ci/ 1 "- +c 2 / 2 - +•••
This linear system can be written in matrix form:
/ h h ••• fn \ / c i \
C2
l / l J 2 ' ' ' in 0.
y An-l) An-1) _ _ _ f^ n_1) )
Vc /
n

By 3.3.2, if the determinant of the coefficient matrix of the
linear system is not identically equal to zero in [a, b], the