Question

Let v1= [−3 0 6]T , v2= [−2 2 3]T , v3= [0 − 6 3]T , and w= [1 14 9]T . (1). Determine if w is in the subspace spanned by v1, v2, v3. (2). Are the vectors v1, v2, v3 linearly dependent or independent? Justify your answer.

Answer 1

Q. 2. Let A = [v₁,v₂,v₃,w] =

-3	-2	0	1
0	2	-6	14
6	3	3	9

To determine whether w is in the subspace spanned by v₁,v₂,v₃ and whether v₁,v₂,v₃ are linearly independent , we will reduce A to its RREF as under:

Multiply the 1st row by -1/3

Add -6 times the 1st row to the 3rd row

Multiply the 2nd row by 1/2

Add 1 times the 2nd row to the 3rd row

Multiply the 3rd row by 1/18

Add -7 times the 3rd row to the 2nd row

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

Add -2/3 times the 2nd row to the 1st row

Then the RREF of A is

1	0	2	0
0	1	-3	0
0	0	0	1

This implies that:

(1). w cannot be expressed as a linear combination of v₁,v₂,v₃ . Therefore, w is not in the subspace spanned by v₁,v₂,v₃ .

(2). The vectors v₁,v₂,v₃ are not linearly independent as v₃ = 2v₁-3v₂ so that 2v₁-3v₂ - v₃ = 0