Find all eigenvalues and a base B formed by eigenvectors of the linear transformation 1: P3 + P3 given byT(f(x)) = af'(x) + f"(x) – f(2)

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Answered: Find all eigenvalues and a base B…

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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3 *. Sketch a graph of the eigenvectors and plot Solve the system of ODES given by i' = | , some sample trajectories. Is the origin an attractor, a repeller or a saddle point? -2.
9. Let z = x + iy with x, Y E R and thus z = x – iy. Consider the maps iy + (; ?) - atr.v), 1++ (;). i+(!) = A(x, y), 1++ ()- x + iy + (i) Calculate z? and A2(x,y). Discuss. (ii) Find the eigenvalues and normalized eigenvectors of A(x, y). (iii) Calculate A(x,y) ® A(x,y) and find the eigenvalues and normalized eigen- vectors.
Let x(t) x₁ (t) x' (t) x₁ (t): = = If x (0) = x₂(t): = = = ] Put the eigenvalues in ascending order when you enter x₁ (t), x₂(t) below. exp( t) + exp( x₁(t) x₂(t) -27 x₁(t) + 12x₂(t) -56 x₁(t) + 25 x₂(t) 4 be a solution to the system of differential equations: -2 exp( find x(t). t) + exp( t) t)
5. Let A be a 6 x 6 matrix with characteristic equation X2(a – 1) (A – 2)3= 0. What are the possible dimensions for eigenspaces of A?
Consider the following interpolating points for the function f(x) = x² – 2x: Xi 01 23 f(xi) 0 -1 03 (a) Use monomial basis to find P₁(x) that interpolates f(x) at x; for i = 0,1. Con- struct the Vandermonde matrix and then use the backslash \ in matlab to solve the linear system. (b) Repeat (a) to find P₂(x) that interpolates f(x) at x; for i = 0, 1, 2. (c) Repeat (a) to find P3(x) that interpolates f(x) at x, for i = 0, 1, 2, 3. (d) Compare P₂(x), P3(x) and f(x). What do you see? Can you explain it?
kindly answer it perfectly i will appreciate your answer
Verify that λ; is an eigenvalue of A and that x; is a corresponding eigenvector. 3 0 A = [ A₁ = 3, x₁ = (1, 0) 2₂ = -3, x₂ = (0, 1) 0 -3 AX1 ~-[D-F= -0) --- 3 = 0 -3 3 *-*«DE÷-0→ AX2 = = = ไป = 22x2 ↓ 1
What are the eigenvalues and eigenfunctions: x′′ + λx = 0, x(1) = x(3), x′(1) = x′(3) - Definey by x(z) = y((z−2)π) for 1 ≤ z ≤ 3. Show that y(−π) = y(π) and y′(−π) = y′(π) - Substitute into the equation to get y′′((z−2)π) + (λ/π2) y((z−2)π), for 1 ≤ z ≤ 3 - Use the change of variable t = (z − 2)π to show that the above equation has a non-zero solution if and only if either λ = k2π2 for some integer k ≥ 1 or λ = 0 and the solutions (eigenfunctions) are given by cos(kt), sin(kt) and 1 for −π ≤ t ≤ π. - Plug back t = (z − 2)π to find the eigenfunctions and eigenvalues of original equation
Verify that λ; is an eigenvalue of A and that x; is a corresponding eigenvector. 2₁9, x₁ (1, 0) = - [8. AX1 = AX2 A = = 9 0 0-9 = 22-9, x₂ = (0, 1) = = ↓ 1 = 9 0 -9 22x2 ~-(:D)- E= -0)-*-*2 1 0-9 ↓1 •[:] · = 2₁x1
The value of 'a' for whichf: R R, ax + 7 flx) is invertible lies in x* + 4 (a) (0, 2) (b) (1, 0) (c) (0, 0) (d) none of these
Q2
Consider the quadratic form Q(x, y) = -² A = la cla 14 9 ala 10 Ila 28 9 ---/- 14 9 14 9 14 9 20 9 56 x y 9 following principal minors of the matrix A: The (leading) principal minor 1pm3 = pm3 of order 3: The leading principal minor 1pm² = pm3 of order 2: The leading principal minor 1pm₁ = pm of order 1: The principal minor pm² of order 2: The principal minor pm² of order 2: The principal minor pm of order 1: The principal minor pm of order 1: 9 Now use your answers to classify Q: (No answer given) -28 x z We want to classify this using Sylvester's Criteria (the principal minor test). Calculate the 28 y z 9 + 202² with symmetric coefficient matrix

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Find all eigenvalues and a base B formed by eigenvectors of the linear transformation 1: P3 + P3 given by T(f(x)) = af'(x) + f"(x) – f(2)