Homework SourseLet F11 1 1 1 0 F12 0 1 1 1 F21 0 0 0 1 and F22 1 1 1 1
Let F_11: = [1 1 -1 0] F_12: = [0 -1 1 1], F_21: = [0 0 0 1], and F_22: = [1 -1 1 -1] Show that the matrices {F_11, F_22- F_21, F_22} form a basis of 2 times 2 matrices by expressing the matrices E_11: = [1 0 0 0], E_12: = [0 0 1 0], E_21: = [0 1 0 0], and F_22: = [0 0 0 1] as linear combinations of the matrices F_11, F_12, F_21, and F_22.
![Let F_11: = [1 1 -1 0] F_12: = [0 -1 1 1], F_21: = [0 0 0 1], and F_22: = [1 -1 1 -1] Show that the matrices {F_11, F_22- F_21, F_22} form a basis of 2 times 2](/WebImages/37/let-f11-1-1-1-0-f12-0-1-1-1-f21-0-0-0-1-and-f22-1-1-1-1-1110207-1761588511-0.webp "Let F_11: = [1 1 -1 0] F_12: = [0 -1 1 1], F_21: = [0 0 0 1], and F_22: = [1 -1 1 -1] Show that the matrices {F_11, F_22- F_21, F_22} form a basis of 2 times 2")
![Let F_11: = [1 1 -1 0] F_12: = [0 -1 1 1], F_21: = [0 0 0 1], and F_22: = [1 -1 1 -1] Show that the matrices {F_11, F_22- F_21, F_22} form a basis of 2 times 2](/WebImages/37/let-f11-1-1-1-0-f12-0-1-1-1-f21-0-0-0-1-and-f22-1-1-1-1-1110207-1761588511-1.webp "Let F_11: = [1 1 -1 0] F_12: = [0 -1 1 1], F_21: = [0 0 0 1], and F_22: = [1 -1 1 -1] Show that the matrices {F_11, F_22- F_21, F_22} form a basis of 2 times 2")
Solution
We have F11 = E 1–E2+E3 , F12 = E2 –E3+E4, F21 = E2+E4 and F22 = E1+E2-E3-E4 .
Now, let A =
1
0
0
1
-1
1
1
1
1
-1
0
-1
0
1
1
-1
We will reduce A to its RREF as under:
Add 1 times the 1st row to the 2nd row
Add -1 times the 1st row to the 3rd row
Add 1 times the 2nd row to the 3rd row
Add -1 times the 2nd row to the 4th row
Multiply the 4th row by -1/3
Add -2 times the 4th row to the 2nd row
Add -1 times the 4th row to the 1st row
Add -1 times the 3rd row to the 2nd row
Then the RREF of A is
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
This implies that the matrices {F11 , F12 , F21 , F22} are linearly independent and form a basis for M22,the vector space of all 2x2 matrices.
| 1 | 0 | 0 | 1 |
| -1 | 1 | 1 | 1 |
| 1 | -1 | 0 | -1 |
| 0 | 1 | 1 | -1 |
