Consider the following B 1 1 2 1 B 12 0 4 4 xB 1 3 a Find

Consider the following. B = {(-1, -1), (2, 1)}, B\'= {(-12, 0), (-4, 4)}, [x]_B = [-1 3] (a) Find the transition matrix from B to B\' p^-1 = (b) Find the transition matrix from B\' to B. P = (c) Verify that the two transition matrices are inverses of each other. PP^-1 = (d) Find the coordinate matrix [x]_B, given the coordinate matrix [x]_B.

Solution

(a) The transition matrix from B to B’ can be obtained by row-reducing to its RREF, as under, the matrix

-12

-4

-1

2

0

4

-1

1

         
Multiply the 1st row by -1/12

Multiply the 2nd row by ¼

Add -1/3 times the 2nd row to the 1st row

The RREF is

1

0

1/6

-1/4

0

1

-1/4

1/4

Therefore, the transition matrix from B to B’ is P^-1 =

1/6

-1/4

-1/4

1/4

(b) The transition matrix from B’ to B can be obtained by row-reducing to its RREF, as under, the matrix

-1

2

-12

-4

-1

1

0

4

Multiply the 1st row by -1

Add 1 times the 1st row to the 2nd row

Multiply the 2nd row by -1          

Add 2 times the 2nd row to the 1st row

The RREF is

1

0

-12

-12

0

1

-12

-8

Therefore, the transition matrix from B’ to B is P =

-12

-12

-12

-8

(c ) The inverse of the matrix

-12

-12

-12

-8

is

1/6

-1/4

-1/4

1/4

(d) We have [x]_B’ = (-1,3)^T. Hence [x]_B = P [x]_B’ = (-24,-12)^T.

-12	-4	-1	2
0	4	-1	1