Concept explainers
(Change of Basis) Set
and set
(a) We can use the MATLAB function rank to determine whether the column
(b) Use MATLAB to compute the transition matrix from the standard basis for
(c) Use MATL.AB to compute the transition matrix from the standard basis to the ordered basis
(d) Use MATLAB to compute the transition matrix S from E to F and the transition matrix T from F to E. How are S and T related? Verify that
a.
Compute the rank of the given matrix.
Answer to Problem 1E
The rank of the matrix is
Explanation of Solution
Given information:The matrixes are given
Given,
Using row reduce echelon form to compute the rank of the given matrix
The rank of the matrix is
Program:
clc clear close all U=round(20*rand(4))-10; V=round(10*rand(4)); b=ones(4,1); ru=rank(U); Urow=rref(U) rv=rank(V); Vrow=rref(V)
Query:
- First, we have defined the given matrix.
- Then use the function “rank” to compute the rank of the matrix.
- Get the rank of the given matrix.
b.
Compute the transition matrix of basis of order 4 for the given matrix U.
Answer to Problem 1E
In the comparison we get the answer, which is close to zero.
Explanation of Solution
Given information:The matrixes are given
Given,
Using row reduce echelon form to compute the rank of the given matrix
The transition matrix of the given matrix is
Coordinate of b with respect to ordered basis E are given
Calculate the coordinate vector
To verify use
Then we get
The above vector is close to zero.
Program:
clc clear close all U=round(20*rand(4))-10; V=round(10*rand(4)); b=ones(4,1); ru=rank(U); Urow=rref(U); rv=rank(V); Vrow=rref(V); d=inv(U); c=d*b; b-U*c
Query:
- First, we have defined the given matrix.
- Then use the function “rank” to compute the rank of the matrix.
- Calculate the transition matrix.
- Then use the condition to calculate the coordinate vector.
- Then verify the given relation.
c.
Compute the transition matrix of basis of order 4 for the given matrix V.
Answer to Problem 1E
In the comparison we get the answer, which is close to zero.
Explanation of Solution
Given information:The matrixes are given
Given,
Using row reduce echelon form to compute the rank of the given matrix
The transition matrix of the given matrix is
Coordinate of b with respect to ordered basis E are given
Calculate the coordinate vector
To verify use
Then we get
The above vector is close to zero.
Program:
clc clear close all U=round(20*rand(4))-10; V=round(10*rand(4)); b=ones(4,1); ru=rank(U); Urow=rref(U); rv=rank(V); Vrow=rref(V); d=inv(V); d=d*b; b-V*d
Query:
- First, we have defined the given matrix.
- Then use the function “rank” to compute the rank of the matrix.
- Calculate the transition matrix.
- Then use the condition to calculate the coordinate vector.
- Then verify the given relation.
d.
Show the given relationship between given matrix.
Answer to Problem 1E
The solution is
Explanation of Solution
Given information:The matrixes are given
Given,
Using row reduce echelon form to compute the rank of the given matrix
The transition matrix for E to F is
And, the transition matrix for F to E is
Then show
And, show the given condition
Program:
clc clear close all U=round(20*rand(4))-10; V=round(10*rand(4)); b=ones(4,1); ru=rank(U); Urow=rref(U); rv=rank(V); Vrow=rref(V); c=inv(U)*b; d=inv(V)*b; S=inv(V)*U; T=inv(U)*V; S-inv(T) d-S*c c-T*d
Query:
- First, we have defined the given matrix.
- Then use the function “rank” to compute the rank of the matrix.
- Calculate the transition matrix.
- Then use the condition to calculate the coordinate vector.
- Then verify the given relation.
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