Let V = P2(R) = {ax² + bx +c | a, b, c € R}. With inner product%3D(Si g) = | 5()g(t)dt.Define linear operator T(f) = f'.(i) Find T".(ii) Is T normal?(iii) Is T self-adjoint?(iv) Does T have an orthonormal basis of eigenvectors? If so, find one.

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Answered: Let V = P2(R) = {ax² + br +c a, b, c…

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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V =R * R 10a6b T(") - (170-100) {(2)· (3)} B = { find the matrix representation of T with respect to beta. Then determine wether beta is a basis consisting of eigenvectors. I have got -2 [T]₁ = ( 2 4 this does not match answer key. What's wrong? please point it out and have the rest of solution to the problem. thanks!
Let B = {1,x, r²} be the standard ordered basis for P2(R). Let T be the linear operator on P2(IR) defined by T(f(x)) = 2f'(x) + f(1)x. %3D Find the eigenvalues and eigenvectors of T.
a' = with one of the phase plane direction fields below: A B D (The blue lines are the arrow shafts, and the black dots are the arrow tipa.) Note: To solve this problem, you only need to compute eiguvalus. In fact, it is enough to just compute whether the cigenvalues are real or complex and positive or negative, If you use Trace-Determinant plane, give a brief explanation about eigenvalues.
6. Let A be an n x n hermitian matrix, i.e. A = A*. Assume that all n eigenvalues are different. Then the normalized eigenvectors { v; : j = 1,...,n} form an orthonormal basis in C". Consider B:= (Ax – ux, Ax – vx) = (Ax – ux)*(Ax – vx) | where (, ) denotes the scalar product in C" and µ,v are real constants with µ 0.
Define T(f) = f′ from P4 to P4. (a) Prove T is linear. (b) Give the matrix for T. Is it invertible? Why or why not?
(a) Show that the vector functions are linearly independent on the real line. (b) Why does it follow from Theorem 2 that there is no continuous matrixP(t) such that x1 and x2 are both solutions of x' = P(t)x?
7. Let A be an n x n Hermitian matrix. (a) Show that (Au, v) = (u, Av) for any n-dim vectors u and v. Here (, ·) denotes the inner product. (Hint: rewrite the inner product as matrix multiplication.) (b) Let x be an eigenvector associated to the eigenvalue A. Based on part (a), show that A(x, x) = X(x, x). (c) As in part (b), show that = X; that is, the eigenvalue A is real. 1
Consider the vector space V of all the real polynomials of degree less than or equal to three, p(x) = ax^3 + b x^2 + c x + d, and consider the linear transformation T: V ------> V given by the derivative, i.e., T( p ) = p' = 3ax^2 + 2 b x + c. Find the eigenvalues and eigenvectors of this transformation.
(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.

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Let V = P2(R) = {ax² + br +c a, b, c ER}. With inner product (S, g) = / s(t)g(t)dt. Define linear operator T(f) = f' . (i) Find T. (ii) Is T normal? (iii) Is T self-adjoint? (iv) Does T have an orthonormal basis of eigenvectors? If so, find one.