Are the various v1 1 1 1T v2 1 2 2T and v3 2 1 1T linearly

Are the various v_1 = (1, -1, 1)^T, v_2 = (1, 2, -2)^T, and v_3 =(2, 1, -1)^T linearly independent? What is the dimension of span {v_1, v_2, v_3}? Explain your answers.

Solution

If v₁,v₂,v₃ are linearly independent, none of these will be a linear combination of the others. In the case of the contrary being true, these vectors will be linearly dependent. In order to ascertain this, let us create a matrix A with these vectors as columns. Then A =

1

1

2

-1

2

1

1

-2

-1

We will reduce A to its RREf as under:

Add 1 times the 1st row to the 2nd row

Add -1 times the 1st row to the 3rd row

Multiply the 2nd row by 1/3

Add 3 times the 2nd row to the 3rd row

Add -1 times the 2nd row to the 1st row

Then the RREF of A is

1

0

1

0

1

1

0

0

0

Then, apparently, v₃=v₁+v₂. Hence, v₁,v₂,v₃ are linearly dependent.Also, the dimension of span { v₁,v₂,v₃ } is 2.

1	1	2
-1	2	1
1	-2	-1