Let T: R³ R³ be the linear transformation that does the following things, in this order, to an input vector x= [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take x = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AT) is geometrically.
Let T: R³ R³ be the linear transformation that does the following things, in this order, to an input vector x= [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take x = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AT) is geometrically.
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
![2. Let T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector
x = [x y z]¹:
(i) Interchanges the second and third coordinates of .
(ii) Multiplies the first coordinate of the resulting vector by 2.
(iii) Replaces the second coordinate of the resulting vector with a 0.
(iv) Multiplies the resulting vector by the following matrix:
0
0
0
0
You don't have to show that T is linear.
(a) The description of T given above is purely algebraic, in that it explicitly describes how to take =
[x y z] and write down T() in coordinates.
Give a geometric description of what each of the four "steps" of applying T actually does to a vector.
(Your Week 9 tutorials may help in describing what the last step does.)
(b) Find the standard matrix AT of T.
(c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet-
rically as a subset of R³)
(d) Find a spanning set for im(AT), and describe what im(AȚ) is geometrically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc84b9504-04d8-4492-b13a-660d2a574b53%2F499c9982-3e28-41f0-925b-675718e9f2a1%2Fy9oq737_processed.png&w=3840&q=75)
Transcribed Image Text:2. Let T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector
x = [x y z]¹:
(i) Interchanges the second and third coordinates of .
(ii) Multiplies the first coordinate of the resulting vector by 2.
(iii) Replaces the second coordinate of the resulting vector with a 0.
(iv) Multiplies the resulting vector by the following matrix:
0
0
0
0
You don't have to show that T is linear.
(a) The description of T given above is purely algebraic, in that it explicitly describes how to take =
[x y z] and write down T() in coordinates.
Give a geometric description of what each of the four "steps" of applying T actually does to a vector.
(Your Week 9 tutorials may help in describing what the last step does.)
(b) Find the standard matrix AT of T.
(c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet-
rically as a subset of R³)
(d) Find a spanning set for im(AT), and describe what im(AȚ) is geometrically.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 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,
