Answer :
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:
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