Homework SourseLet B 1 3 2 2 and B 12 0 4 4 be bases for R2 and let A be
Let
B = {(1, 3), (?2, ?2)} and B\' = {(?12, 0), (?4, 4)} be bases for R2,
and let
A =
be the matrix for
T: R2 ? R2 relative to B.
(a) Find the transition matrix P from B\' to B.
(b) Use the matrices P and A to find [v]B and [T(v)]B, where
[v]B\' = [2 ?1]T.
[v]B
[T(v)]B
(c) Find P?1 and A\' (the matrix for T relative to B\').
(d) Find
[T(v)]B\'
two ways.
[T(v)]B\' = P?1[T(v)]B
[T(v)]B\' = A\'[v]B\'
| 0 | 2 |
| ||
| 3 | 4 |
![\"leftbracket2.gif\"](\"//www.webassign.net/wastatic/wacache3b8a614226a953a8cd9526fca6fe9ba5/watex/img/leftbracket2.gif\"/)
![\"rightbracket2.gif\"](\"//www.webassign.net/wastatic/wacache3b8a614226a953a8cd9526fca6fe9ba5/watex/img/rightbracket2.gif\"/)
Solution
(a) Let M =
1
-2
-12
-4
3
-2
0
4
The RREF of M is
1
0
6
4
0
1
9
4
Hence, the transition matrix P from B\' to B is P =
6
4
9
4
(b). Since [v]B’ = (2 -1)T, hence [v]B = P[v]B’ = (8,14)T.
( c). Let N = [P|I2] =
6
4
1
0
9
4
0
1
The RREF of N is
1
0
-1/3
1/3
0
1
¾
-1/2
Hence, P-1 =
-1/3
1/3
¾
-1/2
Also, A’ = PA =
12
28
12
34
(d). [T(v)]B’ = A’[v]B’ = (-4,10)T
| 1 | -2 | -12 | -4 |
| 3 | -2 | 0 | 4 |
