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Let B = {(1, 3), (−2, −2)} and B' = {(−12, 0), (−4, 4)} be bases for R2, and let A = 0 2 3 4 be the matrix for T: R2 → R2 relative to B.

(a) Find the transition matrix P from B' to B. P =

(b) Use the matrices P and A to find [v]B and [T(v)]B, where [v]B' = [−2 4]T. [v]B = [T(v)]B =

(c) Find P−1 and A' (the matrix for T relative to B'). P−1 = A' = (

(d) Find [T(v)]B' two ways. [T(v)]B' = P−1[T(v)]B = [T(v)]B' = A'[v]B' =

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Respuesta :

fortuneonyemuwa

Answer:

Step-by-step explanation:

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

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