Homework SourseWhich of the following sets of polynomials form a basis for
Which of the following sets of polynomials form a basis for P2?
| (i) p1 = 1 + x, p2 = 6x + x2, p3 = 1 + 6x + x2 |
| (ii) p1 = 1, p2 = 1 2x, p3 = x, p4 = x2 |
| (iii) p1 = 1 + 4x + 5x2, p2 = 4 + x |
Solution
(i) Let A =
1
0
1
1
6
6
0
1
1
The RREF of A is I3. This implies that p1,p2,p3 are linearly independent and form a basis for P2.
(ii) Let A =
1
1
0
0
0
-2
1
0
0
0
0
1
The RREF of A is
1
0
1/ 2
0
0
1
-1/2
0
0
0
0
1
This implies that p3 = p1/2 –p2/2 so that p1,p2,p3,p4 are linearly dependent and , therefore, do not form a basis for P2. However, p1,p2,p4 are linearly independent and form a basis for P2.
(iii) Let A =
1
4
4
1
5
0
The RREF of A is
1
0
0
1
0
0
This implies that athough p1,p2 are linearly independent, they do not form a basis for P2 as the RREF of A does not have 1 in the last row.
| 1 | 0 | 1 |
| 1 | 6 | 6 |
| 0 | 1 | 1 |
