Let T : R3onto the plane x2 =→ R³ be the transformation that projects each vector x =0, that is, T(x1, x2, X3) = (x1,0, x3).(x1, x2, X3)(a) Show that T is a linear transformation.(b) Find the matrix representation of T.(c) Is T one-to-one? Is T onto? Justify your answers.

Given transformation is T:R3→R3 such that T(x1,x2,x3)=(x1,0,x3) a) show that T is a linear…

Answered: Let T : R3 → R³ be the transformation…

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…

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Let T : R3 → R³ be the transformation that projects each vector x = (x1, X2, X3) onto the plane x2 = 0, that is, T(¤1, T2, T3) = (x1,0, x3). %3D (a) Show that T is a linear transformation. (b) Find the matrix representation of T. (c) Is T one-to-one? Is T onto? Justify your answers.