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 7 = [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 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 7 = [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
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![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 7 =
[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%2F9a487e32-6445-4eef-9c62-953e844751ce%2Fee342256-6897-43c3-8a61-c36bb6c77c0e%2Fvty4s3u_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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 7 =
[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.
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