Homework SourseLet upsilon1 1 2 3 2 upsilon2 2 2 1 1 upsilon3 4 2 3 1 an
Solution
2. Let A =
-1
2
-4
-3
0
2
-2
2
-1
-1
-3
1
3
3
-3
-2
1
1
-1
-4
We will reduce A to its RREF as under:
Multiply the 1st row by -1 ; Add -2 times the 1st row to the 2nd row
Add 3 times the 1st row to the 3rd row ; Add 2 times the 1st row to the 4th row
Multiply the 2nd row by ½ ; Add 5 times the 2nd row to the 3rd row
Add 3 times the 2nd row to the 4th row; Multiply the 3rd row by -2/11
Add 11/2 times the 3rd row to the 4th row ; Add 7/2 times the 3rd row to the 2nd row
Add -3 times the 3rd row to the 1st row ; Add 2 times the 2nd row to the 1st row
Then the RREF of A is
1
0
-2
0
3
0
1
-3
0
3
0
0
0
1
1
0
0
0
0
0
Apparently, v3 = -2v1-3v2. Further, 2v1 +3v2+v3+ 0v4 = 0 and since, (0,-1,-3,-4)T= 3v1 +3v2 +v4, hence this vector is in span{S}.
3. Let B =
-2
-1
6
1
2
-1
1
3
1
-2
-1
2
3
-2
6
3
3
-9
-2
-3
We will reduce B to its RREF as under:
Multiply the 1st row by -1/2 ; Add 1 times the 1st row to the 2nd row
Add 1 times the 1st row to the 3rd row ; Add -3 times the 1st row to the 4th row
Multiply the 2nd row by 2/3 ; Add -5/2 times the 2nd row to the 3rd row
Add -3/2 times the 2nd row to the 4th row ; Multiply the 3rd row by -3/10
Add 1 times the 3rd row to the 4th row ; Add -1/3 times the 3rd row to the 2nd row
Add 1/2 times the 3rd row to the 1st row ; Add -1/2 times the 2nd row to the 1st row
Then the RREF of B is
1
0
-3
0
-2
0
1
0
0
-1
0
0
0
1
-3
0
0
0
0
0
Apparently, only (-2, -1,-1,3)T, (-1,1,2,3)T and ( 1,1,-2,-2)Tare linearly independent and ( 6,3,3,-9)T and ( 2,-2,6,-3)Tare linear combinations of (-2, -1,-1,3)T, (-1,1,2,3)T and ( 1,1,-2,-2)T.
A basis for W is { (-2, -1,-1,3)T, (-1,1,2,3)T , ( 1,1,-2,-2)T}. Further, dim (W) = 3.
| -1 | 2 | -4 | -3 | 0 |
| 2 | -2 | 2 | -1 | -1 |
| -3 | 1 | 3 | 3 | -3 |
| -2 | 1 | 1 | -1 | -4 |
![Let upsilon_1 = [-1 2 -3 -2], upsilon_2 = [2 -2 1 1], upsilon_3 = [-4 2 3 1], and upsilon_4 = [-3 -1 3 -1] the set {upsilon_1, upsilon_2, upsilon_3, upsilon_4}](/WebImages/34/let-upsilon1-1-2-3-2-upsilon2-2-2-1-1-upsilon3-4-2-3-1-an-1101614-1761582169-0.webp "Let upsilon_1 = [-1 2 -3 -2], upsilon_2 = [2 -2 1 1], upsilon_3 = [-4 2 3 1], and upsilon_4 = [-3 -1 3 -1] the set {upsilon_1, upsilon_2, upsilon_3, upsilon_4}")
![Let upsilon_1 = [-1 2 -3 -2], upsilon_2 = [2 -2 1 1], upsilon_3 = [-4 2 3 1], and upsilon_4 = [-3 -1 3 -1] the set {upsilon_1, upsilon_2, upsilon_3, upsilon_4}](/WebImages/34/let-upsilon1-1-2-3-2-upsilon2-2-2-1-1-upsilon3-4-2-3-1-an-1101614-1761582169-1.webp "Let upsilon_1 = [-1 2 -3 -2], upsilon_2 = [2 -2 1 1], upsilon_3 = [-4 2 3 1], and upsilon_4 = [-3 -1 3 -1] the set {upsilon_1, upsilon_2, upsilon_3, upsilon_4}")
![Let upsilon_1 = [-1 2 -3 -2], upsilon_2 = [2 -2 1 1], upsilon_3 = [-4 2 3 1], and upsilon_4 = [-3 -1 3 -1] the set {upsilon_1, upsilon_2, upsilon_3, upsilon_4}](/WebImages/34/let-upsilon1-1-2-3-2-upsilon2-2-2-1-1-upsilon3-4-2-3-1-an-1101614-1761582169-2.webp "Let upsilon_1 = [-1 2 -3 -2], upsilon_2 = [2 -2 1 1], upsilon_3 = [-4 2 3 1], and upsilon_4 = [-3 -1 3 -1] the set {upsilon_1, upsilon_2, upsilon_3, upsilon_4}")
![Let upsilon_1 = [-1 2 -3 -2], upsilon_2 = [2 -2 1 1], upsilon_3 = [-4 2 3 1], and upsilon_4 = [-3 -1 3 -1] the set {upsilon_1, upsilon_2, upsilon_3, upsilon_4}](/WebImages/34/let-upsilon1-1-2-3-2-upsilon2-2-2-1-1-upsilon3-4-2-3-1-an-1101614-1761582169-3.webp "Let upsilon_1 = [-1 2 -3 -2], upsilon_2 = [2 -2 1 1], upsilon_3 = [-4 2 3 1], and upsilon_4 = [-3 -1 3 -1] the set {upsilon_1, upsilon_2, upsilon_3, upsilon_4}")
