Consider the ordered bases B = ((3, 5), (1, 2)) and C = ((0, 1), (-3, -2)) for t
ID: 3112997 • Letter: C
Question
Consider the ordered bases B = ((3, 5), (1, 2)) and C = ((0, 1), (-3, -2)) for the vector space R^2. a. Find the transition matrix from C to the standard ordered basis E = ((1, 0), (0, 1)). T^C_E = b. Find the transition matrix from B to E. T^B_E = c. Find the transition matrix from E to B. T_E^B = d. Find the transition matrix from C to B T_C^B e. Find the coordinates of u = (-1, 3) in the ordered basis B. Note that [u]_B = T_E^B [u]_E. [u]_B = f. Find the coordinates of v in the ordered basis B if the coordinate vector of v in C is [v]_C = (1, -2). [v]_B =Explanation / Answer
a.The transition matrix from the ordered basis C for the vector space R2 to the standard ordered basis E i.e. TCE is
0
-3
1
-2
b. The transition matrix from the ordered basis B for the vector space R2 to the standard ordered basis E i.e. TBE is
3
1
5
2
c. Let A =
3
1
1
0
5
2
0
1
To determine the transition matrix from the standard basis E for the vector space R2 to the ordered basis B i.e. TEB , we will reduce A to its RREF as under:
Multiply the 1st row by 1/3
Add -5 times the 1st row to the 2nd row
Multiply the 2nd row by 3
Add -1/3 times the 2nd row to the 1st row
Then the RREF of A is
1
0
2
-1
0
1
-5
3
Thus TEB =
2
-1
-5
3
d. Let A =
3
1
0
-3
5
2
1
-2
To determine the transition matrix from the ordered basis C for the vector space R2 to the ordered basis B i.e. TCB , we will reduce A to its RREF as under:
Multiply the 1st row by 1/3
Add -5 times the 1st row to the 2nd row
Multiply the 2nd row by 3
Add -1/3 times the 2nd row to the 1st row
Then the RREF of A is
1
0
-1
-4
0
1
3
9
Thus, TCB =
-1
-4
3
9
e. The co-ordinates of the vector u= (-1,3) in the ordered basis B i.e.[u]B = TEB [u]E= (-5,14)
f. The co-ordinates of the vector v in the ordered basis C is (1,-2).Then the co-ordinates of the vector v in the ordered basis B i.e. [v]B = TCB [v]C = (7,-15).
0
-3
1
-2
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