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
Elementary & Intermediate Algebra
Pathways To Math Literacy (looseleaf)
College Algebra (Collegiate Math)
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
University Calculus
Elementary Statistics: A Step By Step Approach
- eric pez Xte in z= Therefore, we have (x, y, z)=(3.0000, 83.6.1 Exercise Gauss-Seidel iteration with Start with (x, y, z) = (0, 0, 0). Use the convergent Jacobi i Tol=10 to solve the following systems: 1. 5x-y+z = 10 2x-8y-z=11 -x+y+4z=3 iteration (x Assi 2 Assi 3. 4. x-5y-z=-8 4x-y- z=13 2x - y-6z=-2 4x y + z = 7 4x-8y + z = -21 -2x+ y +5z = 15 4x + y - z=13 2x - y-6z=-2 x-5y- z=-8 realme Shot on realme C30 2025.01.31 22:35 farrow_forwardUse Pascal's triangle to expand the binomial (6m+2)^2arrow_forwardListen A falling object travels a distance given by the formula d = 6t + 9t2 where d is in feet and t is the time in seconds. How many seconds will it take for the object to travel 112 feet? Round answer to 2 decimal places. (Write the number, not the units). Your Answer:arrow_forward
- Solve by the quadratic formula or completing the square to obtain exact solutions. 2 e 104 OA) -16±3√6 B) 8±√10 O c) -8±√10 OD) 8±3√√6 Uarrow_forwardQuestion 14 (1 point) Listen The frame on a picture is 18 in by 22 in outside and is of uniform width. Using algebraic methods, what is the width of the frame if the inner area of the picture shown is 250 in²2? Write answer to 2 decimal places. (Write the number with no units). 18 in Your Answer: 22 inarrow_forward◄ Listen A vacant lot is being converted into a community garden. The garden and a walkway around its perimeter have an area of 560 square feet. Find the width of the walkway (x) if the garden measures 15 feet wide by 19 feet long. Write answer to 2 decimal places. (Write the number without units). X 15 feet Your Answer: 19 feet Xarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,