State with reasons whether each of the following maps T : P₂(R) → P₂(R) is linear. For thosemaps that are linear, find their matrix with respect to the basis 1, x, x² of P₂ (R).2(a) T(f)(x) = f(x+1), (b) T(ƒ)(x) = f(x) +1.

The defining characteristic of a linear transformation T:V→W is that, for any vectors v1 and v2 in V…

Answered: State with reasons whether each of 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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Find the standard matrices A and A' for T = T2 ∘ T1 and T' = T1 ∘ T2.T1: R2 → R2, T1(x, y) = (x − 5y, 2x + 3y)T2: R2 → R2, T2(x, y) = (0, x)
Let T: R? → R³ be the linear map given by [21 + 7x2 xi - 22 *i + x2 Write does the matrix A associated with T. Is A singular?
Let f(u, v) = (u - v, 1, u + v) and g(x, y, z) = xyz. Find the entry a12 (i.e. the entry in the first row and second column) of the derivative matrices Df(u, v), Dg(x, y, z) and D(gof)(0, 1). (a) a 12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(gof)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) D(gof)(u, v) is a 1 x 2 matrix. D(gof)(u, v) is a 2 x 2 matrix. D(gof)(u, v) is a 3 x 2 matrix. D(gof)(u, v) is a 2 x 3 matrix. D(gof)(u, v) is a real-valued function of u and v. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is not defined. D(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 3 x 2 matrix. D(fog)(x, y, z) is a 2 x 2 matrix. D(fog)(x, y, z) is a 2 x 1 matrix.
Let d : P5(R) → P5(R) denotes the first derivative. Show that d is a linear map. Choose a basis for P5(R) and find the matrix M(d). Use the matrix M(d) to find the derivative of f(x) = −9x^5 + 11x^4 −4x^2 +5x−1.
Consider the function T : R → R², defined by T(r, y, 2) = (x – 2y + 2,1 – 2) (a) Write the matrix for T. (b) For the vectors u = (2,3, – 1), v = (-1,3,5) € R*, verify that T(3u + 4v) = 3T(u) + 4T(v). (c) For the unit vectors i = (1,0, 0), j = (0,1,0), k = (0,0, 1), write the matrix [T] = [T(i) T(j) T(k)]. (i.e. write the matrix [T] whose columns are the vectors T(i), T(j), T(k)) %3D

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State with reasons whether each of the following maps T: P2 (R) → P₂(R) is linear. For those maps that are linear, find their matrix with respect to the basis 1, x, x² of P₂ (R). (a) T(f)(x) = f(x+1), (b) T(ƒ)(x) = f(x) + 1.