This question concerns the following exercise. Exercise Consider the linear transformation t: R³ R³ (x, y, z) → (2x, y + z, z-x). (i) Write down the matrix A of t with respect to the standard basis for R3 for both the domain and codomain. (ii) Determine the matrix B of t with respect to the basis {(1,0, 1), (0, -1,-1), (1, 1, 0)} for both the domain and codomain. (iii) State whether there exists a vector v in R3 with the two properties ve Kert and ve Imt, and give a brief reason for your answer. (a) Explain why the following solution to this exercise is incorrect and/or incomplete, identifying one error or significant omission in each of parts (i)-(iii). For each error or omission, explain the mistake that the writer of the solution has made. (There may be more than one error or omission in each part, but you need identify only one. It should not be a statement or omission that follows logically from an earlier error or omission.) Solution (incorrect and/or incomplete!) 20 -1 01 0 0 1 1 (i) A = (ii) We have t(1, 0, 1) = (2, 1, 0), t(0, -1, -1) = (0, -2, -1) and t(1, 1,0) = (2, 1, -1), so B = /2 0 2 1-2 1 0-1 -1 . (iii) There is no such vector v. This is because Kert is a subset of the domain of t and Imt is a subset of the codomain of t. (b) Write out a correct solution to the exercise.
This question concerns the following exercise. Exercise Consider the linear transformation t: R³ R³ (x, y, z) → (2x, y + z, z-x). (i) Write down the matrix A of t with respect to the standard basis for R3 for both the domain and codomain. (ii) Determine the matrix B of t with respect to the basis {(1,0, 1), (0, -1,-1), (1, 1, 0)} for both the domain and codomain. (iii) State whether there exists a vector v in R3 with the two properties ve Kert and ve Imt, and give a brief reason for your answer. (a) Explain why the following solution to this exercise is incorrect and/or incomplete, identifying one error or significant omission in each of parts (i)-(iii). For each error or omission, explain the mistake that the writer of the solution has made. (There may be more than one error or omission in each part, but you need identify only one. It should not be a statement or omission that follows logically from an earlier error or omission.) Solution (incorrect and/or incomplete!) 20 -1 01 0 0 1 1 (i) A = (ii) We have t(1, 0, 1) = (2, 1, 0), t(0, -1, -1) = (0, -2, -1) and t(1, 1,0) = (2, 1, -1), so B = /2 0 2 1-2 1 0-1 -1 . (iii) There is no such vector v. This is because Kert is a subset of the domain of t and Imt is a subset of the codomain of t. (b) Write out a correct solution to the exercise.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Why is the given solution incorrect and what is the correct solution.
Please.
Transcribed Image Text:This question concerns the following exercise.
Exercise
Consider the linear transformation
R³
t: R³
(x, y, z)
(2x,y+z, z-x).
(i) Write down the matrix A of t with respect to the standard basis
for R3 for both the domain and codomain.
(ii) Determine the matrix B of t with respect to the
basis {(1, 0, 1), (0, -1,-1), (1, 1, 0)} for both the domain
and codomain.
(iii) State whether there exists a vector v in R3 with the two properties
ve Kert and ve Imt,
and give a brief reason for your answer.
(a) Explain why the following solution to this exercise is incorrect and/or
incomplete, identifying one error or significant omission in each of
parts (i)-(iii). For each error or omission, explain the mistake that
the writer of the solution has made.
(There may be more than one error or omission in each part, but you
need identify only one. It should not be a statement or omission that
follows logically from an earlier error or omission.)
Solution (incorrect and/or incomplete!)
20
01
(i) A =
0
0 1 1
(ii) We have t(1, 0, 1) = (2, 1, 0), t(0, -1, -1) = (0, -2, -1)
and t(1, 1,0) = (2, 1,-1), so
B =
2 0 2
1 -2 1
0 -1 -1
(iii) There is no such vector v. This is because Kert is a subset of
the domain of t and Imt is a subset of the codomain of t.
(b) Write out a correct solution to the exercise.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
