1). i. Let A be the 3x3 matrix with the vectors in S as columns. Then A =
2 |
1 |
-1 |
1 |
5 |
1 |
-1 |
4 |
2 |
To determine whether the vectors in S span R3, we will reduce A to its RREF as under:
Multiply the 1st row by ½
Add -1 times the 1st row to the 2nd row
Add 1 times the 1st row to the 3rd row
Multiply the 2nd row by 2/9
Add -9/2 times the 2nd row to the 3rd row
Add -1/2 times the 2nd row to the 1st row
Then the RREF of A is
1 |
0 |
-2/3 |
0 |
1 |
1/3 |
0 |
0 |
0 |
Now it is apparent that the columns of A , and hence the vectors in S are not linearly independent and that (-1,1,2)T = -(2/3)(2,1,-1)T + (1/30(1,5,4)T. Since dim(R3) = 3, the vectors in S do not span R3. Any vector of the form (a,b,c)T, where c is non-zero,is not in span(S).
ii. Let A be the 3x3 matrix with the vectors in S as columns. Then A =
2 |
1 |
0 |
1 |
1 |
1 |
-1 |
-1 |
2 |
To determine whether the vectors in S span R3, we will reduce A 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
Multiply the 2nd row by 2
Add 1/2 times the 2nd row to the 3rd row
Multiply the 3rd row by 1/3
Add -2 times the 3rd row to the 2nd row
Add -1/2 times the 2nd row to the 1st row
Then the RREF of A is I3.
It implies that the columns of A , and hence the vectors in S are linearly independent.Further, since dim(R3) = 3, the vectors in S span R3.
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