Homework SourseLINEAR ALGEBRA Write each vector as a linear combination of
LINEAR ALGEBRA
Write each vector as a linear combination of the vectors in S. (If not possible, enter IMPOSSIBLE.) S = {(6, -7, 8, 6), (4, 6 -4, 1)} u = (6, 25, -20, -3) u = (_______)s_1 + (________)s_2 v = (57/2, 123/4, -18, 21/2) v = (________)s_1 + (_______)s_2 w = (-6, -17, 29/8, 51/8) w = (_______)s_1 + (________)s_2 z = (10, -9, 11, 37/4) z = (______)s_1 + (_________)s_2Solution
(a) Let A be the matrix with the given vectors in S and the vector u as columns. The RREF of A is
1
0
-1
0
1
3
0
0
0
0
0
0
Hence, u = -1(6,-7,8,6)+3(4,6,-4,1)
(b) Let B be the matrix with the given vectors in S and the vector v as columns. The RREF of B is
1
0
3/4
0
1
6
0
0
0
0
0
0
Hence, v = 3/4(6,-7,8,6)+6(4,6,-4,1).
(c) Let C be the matrix with the given vectors in S and the vector w as columns. The RREF of C is
1
0
0
0
1
0
0
0
1
0
0
0
Thus, w cannot be expressed as a linear combination of the given vectors.
(d) Let D be the matrix with the given vectors in S and the vector z as columns. The RREF of D is
1
0
3/2
0
1
1/4
0
0
0
0
0
0
Hence, z = 3/2(6,-7,8,6)+1/4(4,6,-4,1).
| 1 | 0 | -1 |
| 0 | 1 | 3 |
| 0 | 0 | 0 |
| 0 | 0 | 0 |
