Identify the linear transformation matrix A for the transformation R³ → R² such that it satisfiesand (2) a116,9₁2 = 4, anda13= 2.following: (1) im(A) is spanned by the vector[1/4]=

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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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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)).
The cross product of two vectors in R' is defined by 日-屏 [b1 azb3 – azb2 ai a2 b2 azbı – aıb3 b3 aıb2 – azbı аз Let i . Find the matrix A of the linear transformation from R to R° given by T(7) = j x a. A = || ||
Find a preimage of the vector 4 V= under the linear transformation T given by the matrix -3] = [44] -4 14 That is find the vector u such that T(u) = v. A U₁ ū₂
Let B= (b,, b,, b,} and D= (d,, d,} be bases for vector spaces V and W, respectively. Let T :V- W be a linear transformation with the following properties. T(b,) = - 3d, +2d2, T(b2) = - 4d, + 4d2, T(b3) = 5d2 Find the matrix for T relative to B and D. The matrix for T relative to B and D isO.
Consider the following linear transformation and basis. Find the standard matrix A for the linear transformation. A = Find the transition matrix P from B' to the standard basis B and then find its inverse. P = T: R² R², T(x, y) = (x - 5y, yx), B' = {(1, -2), (0, 3)) p-1 = A' = ↓1 ↓ 1 Find the matrix A' for T relative to the basis B'.
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…
is a) and b) linear transformations. If yes, write down the matrix of the transformation
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dentify the linear transformation matrix A for the transformation R³ → R² such that it satisfies ‘ollowing: (1) im(A) is spanned by the vector. [1/2] and (2) a₁1 : 6, α₁2 = 4, and a13 = 2. =