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' =

Question

Konnor

Mathematics

27 November, 20:17

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' =

+2

Answers (1)

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Small · Answer 1

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