In exercise 11, (a) Find the transition matrix from B to , (b) Find the transiti
ID: 3116000 • Letter: I
Question
In exercise 11, (a) Find the transition matrix from B to , (b) Find the transition matrix from to B, (c) Verify that the two transition matrices are inverses of each other, and (d) find the coordinate matrix , given the coordinate matrix .
In exercise 11, (a) Find the transition matrix from B to B, (b) Find the transition matrix from B to B, (c) Verify that the two transition matrices are inverses of each other, and (d) find the coordinate matrix [xl, given the coordinate matrix B = {(1,0), (1,-1)}Explanation / Answer
11(a). We assume that all the given vectors are column vectors. Let P =
1
1
1
1
1
-1
0
-1
To determine the transition matrix M1 (say) from B to B’, we will reduce P to its RREF as under:
Add -1 times the 1st row to the 2nd row
Multiply the 2nd row by -1/2
Add -1 times the 2nd row to the 1st row
Then the RREF of P is
1
0
½
0
0
1
½
1
Thus, M1, i.e. the transition matrix from B to B’ is
½
0
½
1
(b). Let Q =
1
1
1
1
0
-1
1
-1
To determine the transition matrix M2 (say) from B’ to B, we will reduce Q to its RREF as under:
Multiply the 2nd row by -1
Add -1 times the 2nd row to the 1st row
Then the RREF of Q is
1
0
2
0
0
1
-1
1
Thus, M2, i.e. the transition matrix from B; to B is
2
0
-1
1
(c). It is observed at M2 = M1-1 and M1 = M2-1.
(d). We have [x]B = (2,-2)T. Then [x]B’ = M1. [x]B = (1,-1)T
1
1
1
1
1
-1
0
-1