344 Ordinary Differential Equations
taneous homogeneous system of equations
C1Y11 + C2Y12 + · · · + CnYln = 0
C1Y21 + C2Y22 + · · · + CnY2n = 0C1Ynl + C2Yn2 + · · · + CnYnn = 0
which is equivalent to (3), has a nontrivial solution- a solution in which not
all Ci 's are zero. This system may be rewritten in matrix notation as
(Y11
Y21Yn1or more compactly as
(4)Y12
Y22Yn2Yin) Y2n (c1C2 ) (0)^0
......
...
Ynn Cn 0
Yc=O
where Y is then x n matrix whose jth column is the vector Y1· As we stated
earlier, a homogeneous system, such as system (4), has a nontrivial solution if
and only if det Y = 0. Consequently, we have the following important results:
A set {y 1 , Y2, ... , Yn} of n constant column vectors each of size n x 1 islinearly dependent if and only if det Y = 0 where Y is the n x n matrix
whose jth column is Yj·
Or equivalently, the set {y 1 , Y2, ... , Yn} is linearly independent if and only
if detY -=f. 0.
EXAMPLE 1 Determining Linear Dependence or
Linear IndependenceDetermine whether the set of vectorsis linearly dependent or linearly indep endent.