Begin2.DVI

Definition (Linear dependence and independence of vectors)

Two nonzero vectors A
 and 
Bare said to be linearly dependent if it is possible

to find scalars k 1 , k 2 not both zero, such that the equation

k 1 A
 +k 2 B
 =
 0 (6 .3)

is satisfied. If k 1 = 0 and k 2 = 0 are the only scalars for which the above equation

is satisfied, then the vectors A
 and B
 are said to be linearly independent.

This definition can be interpreted geometrically. If k 1
= 0 ,then equation (6.3)

implies that A
=−k^2

k 1

B
=mB
showing that A
is a scalar multiple of B.
That is, A
and

B
have the same direction and therefore, they are called colinear vectors. If A
and

B
are not colinear, then they are linearly independent (noncolinear). If two nonzero

vectors A
and B
are linearly independent, then any vector C
lying in the plane of

A
and B
can be expressed as a linear combination of the these vectors. Construct

as in figure 6-3 a parallelogram with diagonal C
and sides parallel to the vectors A

and B
when their origins are made to coincide.

Figure 6-3. Vector C
is a linear combination of vectors A
and B
.

Since the vector side

−→

DE is parallel to B
and the vector side

−→

EF is parallel to A,
then

there exists scalars mand nsuch that −→DE =mB
and −→EF =nA.
With vector addition,

C
 =−→DE +−→EF =mB
+nA
 (6 .54)

which shows that C
is a linear combination of the vectors A
and B .