Consider the following.T is the reflection through the origin in R²: T(x, y) = (-x, -y), v = (4, 5).(a) Find the standard matrix A for the linear transformation T.A =(b) Use A to find the image of the vector v.T(v)=

Given that T is the reflection through origin in ℝ2 defined as Tx, y=-x, -y. Also given that v=4, 5.…

Answered: Consider the following. T is the…

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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Assume that T is a linear transformation. Find the standard matrix of T. T: R2→R*, T(e,) = (8, 1, 8, 1), and T(e2) = (-6, 9, 0, 0), where e, = (1,0) and e, = (0,1).
10. Let T be the linear transformation from R3 to R$ that rotates every vector around the y-axis by radians, counterclockwise as viewed from the point (0,5, 0), and afterwards multiplies its length by 4. (a) Find the matrix of transformation T. (b) Calculate T((1, 2, 3)).
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Consider the following. T is the clockwise rotation (0 is negative) of 60° in R2, v = (1, 2). (a) Find the standard matrix A for the linear transformation T. A = (b) Use A to find the image of the vector v. T(v) =
Determine the matrix of the linear transformation T(x, y) = (3x + 5y, 3y – 2x) 5 -2 3 (a) -2 3 (b) (c) 3 3 3 (d) 3 3 -2 a 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 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…

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Consider the following. T is the reflection through the origin in R2: T(x, y) = (-x, -y), v = (4, 5). (a) Find the standard matrix A for the linear transformation T. A = 183 ↓ 1 (b) Use A to find the image of the vector v.