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’ lk 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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