Question

Let B = {(1, 3), (?2, ?2)} and B' = {(?12, 0), (?4, 4)} be bases for R2, and

let A =

3	2
0	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' = [1 ?5]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'=

Answer 1

(a). Let M = [B|B’] =

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 from B' to B is P =

6	4
9	4

(b). [v]_B = P[v]_B’ = (-14,-11)^T. Then, [T(v)]_B = A[v]_B = (-64,-44)^T.

(c ). Let N =

6	4	1	0
9	4	0	1

The RREF of N is

1	0	-1/3	1/3
0	1	¾	-1/2

Therefore, P^-1 =

-1/3	1/3
¾	-1/2

Also, A’ = AP^-1 =

½	0
3	-2

(d).          [T(v)]_B’ = P^-1[T(v)]_B= (20/3,-26)^T