1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 343

The following facts are proven in linear algebra.

If the determinant of A is not zero (if det A -1-0), then there is a unique

solution to the system (2^1 ) Ax = b. In particular, if det A -1-0, the homoge-

neous system Ax= 0 h as only the trivia l solution, x = 0.

If the determinant of A is zero (if det A= 0) , then the system (2') Ax= b

does not have a solution or it has infinitely many nonunique solutions. If

det A = 0, the homogeneous system Ax = 0 has infinitely many nonzero

solutions in addition to the trivial (zero) solution.

DEFINITIONS Linearly Dependent and Linearly Independent

Sets of Vectors

A set of m constant vectors {Y1, Y2, ... , Ym} which are a ll the same

size, say n x 1, is said to be linearly dependent if there exist constants

c 1 , c 2 , ... , Cm at least one of which is nonzero, such that

C1Y1 + C2Y2 + · · · + CmYm = 0.

Otherwise, the set {y 1 , y2, ... , Ym} is said to be linearly independent. Hence,

the set of vectors {y 1 , y 2 , ... , Ym} is linearly independent if

C1Y1 + C2Y2 + · · · + CmYm = 0 implies C1 = C2 = · · · = Cm = 0.

That is, the only way to express the zero vector, 0 , as a linear combination

of linearly independent vectors is for all of the coefficients, ci , to be zero.

Now let us consider a set of n constant column vectors each having n com-
ponents, {Y1, Y2, ... , Yn}· We will let

Y1 = (;::) , Y2 = (~:) , · · .,

Ynl Yn2

(

YY2n in)

Yn =. ·

Ynn

Thus, Yij denotes the ith component of the jth vector in the set. The set

{y 1 , y2, ... , Yn} is linearly dependent if and only if there exist constants
c 1 , c2, ... , Cn not all zero such that

(3)

Hence, the set {Y1, Y2, ... , Yn} is linearly dependent if and only if the simul-