Homework Answers
The null space of A is set of solutions to the equation AS = 0.To solve this equation, we will reduce A to its RREF as under:
Multiply the 1st row by 1/2
Add -6 times the 1st row to the 2nd row
Add -4 times the 1st row to the 3rd row
Multiply the 2nd row by -1/3
Add 9 times the 2nd row to the 3rd row
Multiply the 3rd row by 1/3
Add 1 times the 3rd row to the 2nd row
Add -3/2 times the 3rd row to the 1st row
Add -3/2 times the 2nd row to the 1st row
Then the RREF of A is
|
1 |
-5 |
8 |
0 |
0 |
|
0 |
0 |
0 |
1 |
0 |
|
0 |
0 |
0 |
0 |
1 |
If S = (s1,s2,s3,s4,s5)T, then the equation AS = 0 is equivalent to s1-5s2+8s3 = 0 or, s1 =5s1-8s3, s4=0,and s5= 0. Thus, S = (5s2-8s3,s2,s3,0,0)T = s2(5,1,0,0,0)T+s3(-8,0,1,0,0)T. This is the special solution of the equation AS = 0.. Here s2 and s3 are arbitrary real numbers.
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