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Linear Independence 137
x
2
x
3
1
1
O
x
1
Example 2
The vectors and
1
0
0
0
1
0
give us the equation
which has the unique solution
These vectors are therefore also linearly independent.
c
1
= 0
c
2
= 0
= c
1
+ c
2
1
0
0
0
1
0
0
0
0
This one
t?
Example 2
138 Chapter 6 More Vectors
And now we’ lk at linear dependence.
x
2
x
3
1
3
1
O
x
1
The vectors , , and
1
0
0
0
1
0
3
1
0
give us the equation
which has several solutions, for example and
This means that the vectors are linearly dependent.
c
1
= 0
c
2
= 0
c
3
= 0
= c
1
+ c
2
+ c
3
1
0
0
0
1
0
3
1
0
0
0
0
c
1
= 3
c
2
= 1
c
3
= 1
Example 1
Example 1
Linear Independence 139
Example 2
are similarly linearly dependent because there are several
solutions to the equation
Suppose we have the vectors , , , and
1
0
0
0
1
0
0
0
1
a
1
a
2
a
3
as well as the equation
= c
1
+ c
2
+ c
3
1
0
0
0
1
0
0
0
1
0
0
0
+ c
4
a
1
a
2
a
3
The vectors are linearly dependent because there are several
solutions to the system—
for example, and
c
1
= 0
c
2
= 0
c
3
= 0
c
4
= 0
c
1
= a
1
c
2
= a
2
c
3
= a
3
c
4
= 1
The vectors , , , and
a
1
a
2
a
m
1
0
0
0
1
0
0
0
1
+ + c
m
= c
1
a
1
a
2
a
m
0
0
0
1
0
0
0
1
0
0
0
1
+ c
m+1
+ c
2
Among them is but also
c
1
= 0
c
2
= 0
c
m
= 0
c
m+1
= 0
c
1
= a
1
c
2
= a
2
c
m
= a
m
c
m+1
= 1
Example 2

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