Determine whether the vectors v1 v2 v3 from a basis for R3 L

Determine whether the vectors v_1, v_2, v_3 from a basis for R^3. Let v_1 = (1, 2, 3), v_2 = (0, 3, 2), v_3 = (0, 2, 1). Then S = {v_1, v_2, v_3} is a basis for R^3. If w = (4, 3, 2), find (w)s.

Solution

We know that the vectors in a basis are linearly independent and also span the vector space.

Let A = [v₁,v₂,v₃,w]=

1

0

0

4

2

3

2

3

3

2

1

2

In order to ascertain whether S = { v₁,v₂,v₃} is a basis for R³ and also to determine (w)_S , we will reduce A to its RREF as under:

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

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

Multiply the 2nd row by 1/3       

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

Multiply the 3rd row by -3

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

Then the RREF of A is

1

0

0

4

0

1

0

-15

0

0

1

20

This implies that the vectors v₁,v₂,v₃ are linearly independent and also span R³. Further, w = 4v₁ -15v₂ +20v₃ so that (w)_S = (4,-15,20)

1	0	0	4
2	3	2	3
3	2	1	2