3. Consider the transformation T = reflection across the line y = x.(a) Determine T(1) geometrically. Is e an eigenvector? (Yes or No). If Yes, write the corre-sponding eigenvalue.(b) Determine T(2) geometrically. Is e2 an eigenvector? (Yes or No). If Yes, write the corre-sponding eigenvalue.(c) Find an eigenvector ₁ of T with corresponding eigenvalue λ₁ = 1. Use your geometricreasoning. Note: that means T(7₁) = ₁ so v₁ is fixed by T.(d) Find an eigenvector v2 of T with corresponding eigenvalue A2 = -1. Use your geometricreasoning.

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Answered: Consider the transformation T =…

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 eigenvalues and eigenvectors of the linear transformation T: P₁ → P₁ given by T(ax + b) = (6a – 10b)x+ (2a − 3b). Eigenvalue: A = 2 Eigenvalue: A = Ex: 5 Eigenvector: 5x+ Eigenvector: x+1
find the eigen value of L. and eigen Vector they exist, and discuss the dynamic ofL and Portrait it :- ⒸL (3) = (2) Ⓒ> L (5) - ( X + x +vy) (ii) - 2 X +4 tid) L ( 3 ) = ( 3 39
[21] 11. Let A Find the eigenvalues and corresponding eigenvectors of A. Find a matrix P such that P-1AP is a diagonal matrix.
Part d only
snip
For (a) and (b), 2 -1 [A] = -1 2 (a) What are the eigenvalues of A? (b) For each eigenvalue, find a nonzero eigenvector. [1/3] [1/4] (c) Suppose that the matrix B has the eigenvalues 1 and -6, with eigenvectors and respectively. Calculate Q(t).N(0)'. (Q: fundamental matrix for u’=B.u) (d) What is the solution to u'=B.u with u(0)=| ?
Determine if the statements below are True or False.If it’s True, explain why. If it’s False explain why not, or simply give an exampledemonstrating why it’s false If T : R2 → R2is a linear transformation given by counter-clockwise rotationaround the origin by some angle θ, then T either has no (real) eigenvalues, or ithas eigenvalue 1 (only).
2: Let 10 -1 -1 A = 0. 10 10 -1 -1 10 (a) Find the characteristic polynomial of A. (b) Find the dimension of the eigenspace corresponding to the largest eigenvalue.
A = [ 3² -7] ₁ 1. Let A = (a) By finding the eigenvalues and corresponding eigenvectors, give a diagonalization of A. (b) Without computing directly, find A¹ using the diagonalization from part (a).
(c) Let A = 2 1 0 1 3 1 0 1 3 (i) Use the Gershgorin's circle theorem to determine the bounds of the eigenvalue of A. (ii) Use the power method to find the dominant eigenpair of A. (iii) Use the inverse power method to find the least dominant eigenvalue in magnitude and its corresponding eigenvector.
For the matrix Z below, find the eigenvalues and eigenvectors. a. 1 0 -2 Z = 0 5 (-2 0 4 b. Verify if the function below is a linear transformation or not. T:P: →R defined by T (a + bx + cx³) = [2a – b - i. b+c] ii. L: R →R' defined by L = xy| Given the following transformation L(u)=Au, where u = (x, y) and the standard (or induced) с. matrix A is given as A i. Sketch the triangle connecting the points (2, -2), (2, -5) and (5, -2). Then sketch its transformation image if u represents the poit. ii. Identify the type of transformation shown above?
(Tucker) A growth model for elephants (E) and mice (M) predicts population changes from decade to decade. E3E+M M+2E+ 4M a. Determine the eigenvalues and associated eigenvectors for this system. E b. Suppose that the initial population is p = [] = [5]. Write p as a linear combination of the eigenvectors. c. Use the information in part b to estimate the elephant and mice populations in eight decades.

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