Homework SourseDetermine whether the set S is a basis for V i S 1 2 3 4 0
Determine whether the set S is a basis for V. i) S = {[1 2 3 4], [0 1 2 0], [0 0 0 0], [0 1 0 0]}, V = M_2, 2 (ii) S = {1, 1 + 2x}, V = P_2 (iii) S = {(2, 2, 0), (3, 3, 0), (4, 4, 4)}, V = R^3![Determine whether the set S is a basis for V. i) S = {[1 2 3 4], [0 1 2 0], [0 0 0 0], [0 1 0 0]}, V = M_2, 2 (ii) S = {1, 1 + 2x}, V = P_2 (iii) S = {(2, 2, 0](/WebImages/37/determine-whether-the-set-s-is-a-basis-for-v-i-s-1-2-3-4-0-1110120-1761588448-0.webp "Determine whether the set S is a basis for V. i) S = {[1 2 3 4], [0 1 2 0], [0 0 0 0], [0 1 0 0]}, V = M_2, 2 (ii) S = {1, 1 + 2x}, V = P_2 (iii) S = {(2, 2, 0")
Solution
Multiply the 1st row by ½
Multiply the 2nd row by 1/3
Multiply the 3rd row by ¼
Add -4/3 times the 3rd row to the 2nd row
Add -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 I3. This means that the vectors in S are linearly independent and span R3. Hence S is a basis for R3.
