Consider the ordered bases B = (7 - x, 8 - x) and C = (-x, - (2 + x)) for the ve
ID: 3037452 • Letter: C
Question
Consider the ordered bases B = (7 - x, 8 - x) and C = (-x, - (2 + x)) for the vector space P_2[x]. a. Find the transition matrix from C to the standard ordered basis E = (1, x). T^R_C = b. Find the transition matrix from B to E. T^R_B = c. Find the transition matrix from E to B. T^B_E = d. Find the transition matrix from C to B T^B_C = e. Find the coordinates of p(x) = 2x - 1 in the ordered basis B. [p(x)]_B = f. Find the coordinates of q(x) in the ordered basis B if the coordinate vector of q(x) in C is [q(x)]_C = (-1, -2). [q(x)]_B =Explanation / Answer
(a) The transition matrix from C to the standard basis E is
0
-2
-1
-1
(b) The transition matrix from B to the standard basis E is
7
8
-1
-1
(c) Let A =
7
8
1
0
-1
-1
0
1
The RREF of A is
1
0
-1
-8
0
1
1
7
Hence, the transition matrix from E to B is
-1
-8
1
7
(d) Let A =
7
8
0
-2
-1
-1
-1
-1
The RREF of A is
1
0
8
10
0
1
-7
-9
Hence, the transition matrix from C to B is
8
10
-7
-9
(e ) Let A =
7
8
-1
-1
-1
2
The RREF of A is
1
0
-15
0
1
13
Therefore, the coordinates of p(x) =-1+2x in B are ( -15,13)T.
(f ) The coordinates of q(x) in C are (-1,2)T. Then the coordinates of q(x) in B are determined by multiplying the the transition matrix from C to B ( part (d) above) by this vector. Thus, the required coordinates are (12,-11)T
0
-2
-1
-1
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