31. Let W be the subspace sp(sin 2x, cos 2x) ofthe vector space of all real-valued functionswith domain R, and let B = (sin 2x, cos 2x).Find the matrix representation A for thelinear transformation T: W→ W defined byT(f) = D²(f) + 2D(ƒ) + f.

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Answered: 31. Let W be the subspace sp(sin 2x,…

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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Consider the vector space F of all functions from R to R with pointwise addition and scalar multiplication. (a) Show that the three vectors fi = e“, f2 = xe", f3 = x²e" are linearly independent in F. (b) Consider the subspace H = span{f1, f2, f3}. Let T : H → H be the linear transformation defined by df T(f)= + f. dx Find the matrix of T' with respect to the basis B = {fi, f2, f3} of H (i.e., find 8[T]B). (c) Is there a basis C of H in which c |T|c is diagonal? Explain why or why not.
A function T:VW between vector spaces which satisfies T(a+b) = T(a) + T(b) for all a, b eV must be a linear transformation. Select one O True False
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Let M₂x2 be the vector space of all 2 x 2 matrices and define a linear transformation T: M2x2 → M2x2 by a [id] Describe the kernel of T. C T(A) = A + AT, where A = O No answer text provided. odlo 0 -a ° 0 O [1] :d is a real number} [9] : b is a real number} -b O (1 :d is a real number} (d c is a real number} °41 :d is a real number} : a,b,c,d are real numbers} [9] : b is a real number}
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31. Let W be the subspace sp(sin 2x, cos 2x) of the vector space of all real-valued functions with domain R, and let B = (sin 2x, cos 2x). Find the matrix representation A for the linear transformation T: W→ W defined by T(f) = D²(f) + 2D(f) + f.