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.
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
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 Question 2. (15 pts) Let vi=(-3 0 6)", v2= (-2 2 3]", V3= (0 - 6 37, and w= [1 11 9". (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
Question 2. (15 pts) Let Vi= (-3 0 6)", v2= (-2 2 3)", V3= [0 - 6 3)", and w= [1 14 9)? (1). Determine if w is in the subspace spanned by V1, V2, V3. (2). Are the vectors Vi, V2, V3 linearly dependent or independent? Justify your answer.
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