Homework SourseFor each pair of S1 and S2 determine whether i spanS1 spanS2
For each pair of S1 and S2, determine whether (i) span(S1) span(S2), and (ii) span(S2) span(S1).
(a) S1 = {(1, 0,1,0), (1, 2, 1, 2), (3, 1, 2, 1)}, S2 = {(1, 1, 1, 1), (2, 1, 2, 1), (3, 1, 3,1)}.
(b) S1 = {(3,-2, 5, 1), (4,-1, 1, 2), (-2, 1, 2,3)}, S2 = {(9,-4, 4, 0), (0,-2, 11, 3), (-1,-2,-1, 9)}.
Solution
1
1
3
0
2
1
1
1
2
0
2
1
We will reduce A to its RREF as under:
Add -1 times the 1st row to the 3rd row
Multiply the 2nd row by ½
Add -2 times the 2nd row to the 4th row
Multiply the 3rd row by -1
Add -1/2 times the 3rd row to the 2nd row
Add -3 times the 3rd row to the 1st row
Add -1 times the 2nd row to the 1st row
Then the RREF of A is
1
0
0
0
1
0
0
0
1
0
0
0
Similarly, B =
1
2
3
1
1
1
1
2
3
1
1
1
We will reduce B 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
Add -1 times the 1st row to the 4th row
Multiply the 2nd row by -1
Add 1 times the 2nd row to the 4th row
Add -2 times the 2nd row to the 1st row
Then the RREF of B is
1
0
-1
0
1
2
0
0
0
0
0
0
Apparently, spanS2 span S1 and spanS1 span S2.
(b) Let A and B be defined as in part(a) above. Then A =
3
4
-2
-2
-1
1
5
1
2
1
2
3
The RREF of A is
1
0
0
0
1
0
0
0
1
0
0
0
Also, B =
9
0
-1
-4
-2
-2
4
11
-1
0
3
9
The RREF of B is
1
0
0
0
1
0
0
0
1
0
0
0
Apparently, Span S1 = Span S2 so that SpanS2 Span S1 and SpanS1 Span S2.
| 1 | 1 | 3 |
| 0 | 2 | 1 |
| 1 | 1 | 2 |
| 0 | 2 | 1 |
