1550078481-Ordinary_Differential_Equations__Roberts_

346 Ordinary Differential Equations

linearly independent on an interval, it may or may not be linearly independent
at each point in the interval. For example, the two vector functions

are linearly independent on any interval [a, b], but these two vectors are lin-
early dependent at every point p E [a, b]. The vectors are linearly dependent
at any nonzero p, since for c 1 = p and c2 = -1

The vectors are linearly dependent at p = 0, since for any c1 and c2

C1Y1(0) + C2Y2(0) = C1 (~) + C2 (~) = (~).

Thus, the set {y 1 (x), y 2 (x)} is linearly independent on the interval (-oo, oo ),
yet it is linearly dependent at every point in ( -oo, oo)!

We state the following theorem regarding linear independence for vector
functions.

THEOREM A SUFFICIENT CONDITION FOR LINEAR

INDEPENDENCE OF VECTOR FUNCTIONS

Let {y1 (x), Y2(x ), ... , Yn(x)} be a set of n vector functions of size n x 1

and let Y(x) be the matrix with jth column Yj (x ). If det Y(x) f- 0 for any

x E [a, b], then the set {y 1 (x), Y2(x), ... , Yn(x)} is linearly independent on

the interval [a, b].

The converse of this theorem is false as the previous example illustrates,

since the set of vector functions

is linearly independent on any interval [a, b]; yet, for this set

(

x x

2

det Y = det )

0 0 =^0

for all x E [a, b].