Let V be the subspace of the vector space of continuous functions on R spanned by the functions cos(t) and sin(t).Consider the linear transformation T: V→ V given by(T(f))(t) = f"(t) +2f'(t) +9f(t),for f E V.Find the matrix A associated to T with respect to the basis (cos(t), sin(t)) .A =

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Answered: Let V be the subspace of the vector…

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 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 P = (1,3) and Q = (2,6) be points in the plane. Find a vector-valued function R(t) = R0+tv such that R(t) describes the line through P and Q.
Let A = M (f) be the matrix associated with the linear map ƒ((x,y,z)) = (x-2y,y+z,0) relative to the bases B={u= (1,0,3); v = (0,2,1), w = (2,1,0) } and B'= {u'=(0,0,2); v' = (1,0,3); w'=(1,3,1) } Then the first column of A is: O O 1/2 - 1/2 - 1/2 1/2 -1/2 1/2 - 1/2 1/2 -1/2
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) and D(g. f)(0, 1). (a) a12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(g)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) OD(g. f)(u, v) is a real-valued function of u and v. D(g. f)(u, v) is a 2 x 2 matrix. D(g. f)(u, v) is a 2 x 3 matrix. D(g f)(u, v) is a 3 x 2 matrix. 0 D(g f)(u, v) is a 1 x 2 matrix. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 2 x 1 matrix. D(f g)(x, y, z) is not defined. 0 D(f g)(x, y, z) is a 3 x 2 matrix. D(f g)(x, y, z) is a 2 x 2 matrix.
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
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
7. Let V(C) be the vector space of all continuous complex valued functions on 1 [0, 1]. If f(t), g (1) e V then show that (f (t), g (t)) = [ƒ (t) g (t) dt defines an inner product on V (C).
Let the vector C represents the coefficients of a polynomial pt=c1+c2t+c3t2+c4t3. Express the conditions p0=1, p'0=2, p1=1, p'1=0 in the form AC=b.
(c) Let C be the vector space of continuous real-valued functions on R. If T: C → C is defined so T(f(x)) = |ƒ (x)\, is T a linear transformation? Explain why or why not.
1. Suppose p : R → R° and q :R → R are vector-valued functions. Exercise. Further assume p'(-1) = (1,2, 3) and q'(-1) = (1,1, –2). Let v(t) = p(t) + q(t) be yet another vector-valued function. Then v'(-1) = 2. Exercise. Let f(t) = (1,2,3). Find v(t) so that v(0) (0,0, 0) and v'(t) = f(t). v(t) = 3. Exercise. A curve C is described by the vector-valued function p(t) = (t², 4 – 2t?, t² + 2). Exercise. Find all t-values here p p' are orthogonal. your answers in order from least greatest: t = t = t =

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Let V be the subspace of the vector space of continuous functions on R spanned by the functions cos(t) and sin(t). Consider the linear transformation T: V→ V given by (T(f))(t) = f"() +2f’(t) +9f(t), for f € V. Find the matrix A associated to T with respect to the basis (cos(t), sin(t)) . A =