Find the transition matrix from B toB',the transition matrix fromB'to B, verify that the two transition matrices are inverses of each other, and find the coordinate matrix[x]B',given the coordinate matrix[x]B. B = {(−2, 1), (1, −1)}, B' = {(0, 2), (1, 1)}, [x]B =8 −7T

a. {(5,2), (2,1)

b. {(-1/12, 0), (-1/12, 1/4)

c. Yes

Step-by-step explanation:

You've an incomplete question. However the following answers could be gotten from this sample question;

Consider the following.

B = {(5, 2), (2, 1)},

B' = {(−12, 0), (−4, 4)},

[x]B' = [-1,3]

(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

Also attached is an image solution for answer (b)

The problem is solved using Matlab, code and step by step explanation is provided below.

Matlab Code with Step-by-Step Explanation:

clc  

clear all  

format rat  

% We are given two 2x2 matrices B and B' (transpose) and one transpose coordinate matrix xB' (xBT)

B = [-2 1; 1 -1]  

BT= [0 1; 2 1]  

xBT = [8;-7]  

B =

     -2              1        

      1             -1        

BT =

      0              1        

      2              1        

xBT =

      8        

     -7        

a) Find the transition matrix from B to B'

% First of all, create a augmented matrix B' B

Aug=[BT B]  

Aug =

      0              1             -2              1        

      2              1              1             -1        

% Apply Gauss-Jordan elimination using the function rref() and store the result in variable ans  

ans=rref(Aug)  

ans =

      1              0              3/2           -1        

      0              1             -2              1          

% As you can see the transition matrix B to B' is our answer which is located in the 3rd and 4th column so we will extract it and store it B_BT variable.

B_BT = ans(:,[3 4])  

B_BT =

      3/2           -1        

     -2              1      

b) Find the transition matrix from B' to B

% First of all, create a augmented matrix B B'

Aug2=[B BT]  

Aug2 =

     -2              1              0              1        

      1             -1              2              1        

% Again apply Gauss-Jordan elimination using the function rref() and store the result in variable ans2  

ans2=rref(Aug2)  

ans2 =

      1              0             -2             -2        

      0              1             -4             -3        

% As you can see the transition matrix B' to B is our answer which is located in the 3rd and 4th column so we will extract it and store it BT_T variable.

BT_B = ans2(:,[3 4])  

BT_B =

     -2             -2        

     -4             -3    

c) verify that the two transition matrices are inverses of each other

We know that a 2x2 identity matrix is given by

identity=eye(2)  

identity =

      1              0        

      0              1        

If we multiply the two transition matrices and get an indentity matrix then it means that two transition matrices are inverse of each other.

ans=B_BT*BT_B  

ans =

      1              0        

      0              1            

ans2=BT_B*B_BT  

ans2 =

      1              0        

      0              1  

Hence, we got the identity matrix therefore, the two transition matrices are inverse of each other.

d) find the coordinate matrix XB

% The coordinate matrix XB can be found by multiplying the transpose coordinate matrix (xB') with the transition matrix B' to B (BT_B)

xB = BT_B*xBT

xB =

     -2        

    -11

step-by-step explanation: