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.
Want to see more full solutions like this?
Chapter 3 Solutions
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Additional Math Textbook Solutions
Algebra: Structure And Method, Book 1
Beginning and Intermediate Algebra
College Algebra (6th Edition)
Intermediate Algebra (8th Edition)
College Algebra with Modeling & Visualization (6th Edition)
Elementary and Intermediate Algebra: Concepts and Applications (7th Edition)
- True or false? det(A) is defined only for a square matrix A.arrow_forwardConsider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of B and the column vectors of AB are in the column space of A.arrow_forwardShow that no 22 matrices A and B exist that satisfy the matrix equation. AB-BA=1001.arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning