(1 point) Consider the ordered bases B = (-3, -4), (4, 5)) and C = ((-3, -2), (3
ID: 3116218 • Letter: #
Question
(1 point) Consider the ordered bases B = (-3, -4), (4, 5)) and C = ((-3, -2), (3, -4)) for the vector space R2. a. Find the transition matrix from C to the standard ordered basis E = ((1,0), (0, 1)). TE = b. Find the transition matrix from B to E. TE = -4 c. Find the transition matrix from E to B. T3 = d. Find the transition matrix from C to B. e. Find the coordinates of u = (-1, -1) in the ordered basis B. Note that [u]B = TR[u]e. [u]B = f. Find the coordinates of v in the ordered basis B if the coordinate vector of v in C is [vlc = (-1, 1). [v]B =Explanation / Answer
a.The transtion matrix from C to E is TCE =
-3
3
-2
-4
b. The transtion matrix from B to E is TBE =
-3
4
-4
5
c. The transition matrix from E to B can be obtained by row-reducing to its RREF, as under, the matrix A1=
-3
4
1
0
-4
5
0
1
Multiply the 1st row by -1/3
Add 4 times the 1st row to the 2nd row
Multiply the 2nd row by -3
Add 4/3 times the 2nd row to the 1st row
Then the RREF of A1 is
1
0
5
-4
0
1
4
-3
Then TEB =
5
-4
4
-3
d. The transition matrix from C to B can be obtained by row-reducing to its RREF, as under, the matrix
A2=
-3
4
-3
3
-4
5
-2
-4
Multiply the 1st row by -1/3
Add 4 times the 1st row to the 2nd row
Multiply the 2nd row by -3
Add 4/3 times the 2nd row to the 1st row
Then the RREF of A2 is
1
0
-7
31
0
1
-6
24
Then TCB =
-7
31
-6
24
e. Let A =
-3
4
-1
-4
5
-1
The RREF of A is
1
0
-1
0
1
-1
Then the coordinates of the vector u =(-1,-1) with respect to the ordered basis B are uB =(-1,-1)T.
f.
We have v = -1(-3,-2)+ 1(3,-4)= (3,2)+(3,-4) = (6,-2). Let M =
-3
4
6
-4
5
-2
The RREF of M is
1
0
38
0
1
30
Then [v]B = (38,30)T
-3
3
-2
-4
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