Question

Example 2. Test for linear dependency of the following set of vectors: [1, 0,-1], [2, 1, 3). [-1.0, 0], [1, 0, 1) .​

Answer 1

Narad T.

Jul 9, 2018

Please see the explanation below.

Explanation:

The vectors are

{

(

1

,

2

,

−

1

)

,

(

2

,

4

,

6

)

,

(

0

,

0

,

−

8

)

}

The vectors are independent if

α

⎛

⎜

⎝

1

2

−

1

⎞

⎟

⎠

+

β

⎛

⎜

⎝

2

4

6

⎞

⎟

⎠

+

γ

⎛

⎜

⎝

0

0

−

8

⎞

⎟

⎠

=

⎛

⎜

⎝

0

0

0

⎞

⎟

⎠

Where

α

,

β

,

γ

∈

R

3

has only the trivial solution

α

=

β

=

γ

=

0

Perform a row reduction on the augmented matrix

A

=

⎛

⎜

⎝

1

2

0

∣

0

2

4

0

∣

0

−

1

6

−

8

∣

0

⎞

⎟

⎠

⇔

,

R

2

←

R

2

−

2

R

1

⎛

⎜

⎝

1

2

0

∣

0

0

0

0

∣

0

−

1

6

−

8

∣

0

⎞

⎟

⎠

⇔

,

R

2

↔

R

3

⎛

⎜

⎝

1

2

0

∣

0

−

1

6

−

8

∣

0

0

0

0

∣

0

⎞

⎟

⎠

⇔

,

R

2

←

R

2

+

R

1

⎛

⎜

⎝

1

2

0

∣

0

0

8

−

8

∣

0

0

0

0

∣

0

⎞

⎟

⎠

⇔

,

R

2

←

R

2

8

⎛

⎜

⎝

1

2

0

∣

0

0

1

−

1

∣

0

0

0

0

∣

0

⎞

⎟

⎠

⇔

,

R

1

←

R

1

−

2

R

2

⎛

⎜

⎝

1

0

2

∣

0

0

1

−

1

∣

0

0

0

0

∣

0

⎞

⎟

⎠

Since there is a free variable, the system is not linearly independant

The determinant of the matrix

=

0

Therefore,

⎧

⎪

⎨

⎪

⎩

α

=

−

2

γ

β

=

γ

γ

=

free

So,

−

2

γ

⎛

⎜

⎝

1

2

−

1

⎞

⎟

⎠

+

γ

⎛

⎜

⎝

2

4

6

⎞

⎟

⎠

+

γ

⎛

⎜

⎝

0

0

−

8

⎞

⎟

⎠

=

⎛

⎜

⎝

0

0

0

⎞

⎟

⎠

Hope that this will help!!!

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