Question

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

Answer 1

Answer:

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)

Answer 2

Answer:

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