Page 65 - 4549

possible to pick up such numbers C ,C , ,C not all levels to
1 2 n
the zero and such, that linear combination

C y   C y  0 at x  ( , )a b .
1 1 n n

Implementation of this condition means that even one of the
set functions it is possible to express through other linearly. For
0
example, ifC  ,
n
C
y   y   y    y ,   k ,k  1,2, ,(n  1).
n 1 1 2 2 n 1 n 1 k
C
n
Definition 4.5 If at all x  ( , )a b linear combination
C y   C y  0 only then, when the system of functions
1 1 n n
y , y , , y is named linearly independent on ( , )a b .
1 2 n
In this case none of the set functions it is impossible to
express through other linearly.
For example, such elementary functions as
2
2
y  sin x , y  cos x , y  are linearly dependent on
1
1 2 3
)
,
(  , because of C y  C y  C y  0 at
1 1 2 2 3 3
1
C  1,C  1,C   .
1 2 3
For two functions (n=2) y , y it is possible to write down
1 2
such condition of linear dependence :
y C
2   1  k  const , from where y  ky .
y 1 C 2 2 1

This condition is equation of straight line in coordinates
y , y , from here and the name - linear dependence.
1 2

Example. 4.1 Make sure that a system of functions
n
,
)
1, ,x x 2 , , x is linearly independent on (  .
 We differentiate an identity
63