Given definitions:Definition: Let V be a vector space of dimension n, and let T : V → V be a linear transformation. The determinant of T , denoted by det(T ), is defined to be det([T ]B ), where B is any basis for V .Definition: Let V be a vector space of dimension n. The characteristic polynomial of the linear transformation T : V → V is the polynomial in the variable λ given by det([T − λid]B), where B is any basis of V .Definition: Let V be an n dimensional vector space and let T : V → V be a linear transformation. A basis B = {b1,··· ,bn} of V is called an eigenbasis if all bi’s are eigenvectors of T.For part (i), problem 1 is linked here: https://www.bartleby.com/questions-and-answers/for-the-following-transformations-find-an-eigenvector-using-any-methods-you-can-think-of-including-b/7bf163e6-15fc-4ba0-91ac-24333f461731 2. (a) Is the definition of det (T) well defined? What may go wrong? Prove that this definitionis well-defined. 1(b) Let λ be an eigenvalue of T : V → V and let ở be an eigenvector with eigenvalue λ.Then(T - Aid) = 0.This means that the linear transformation T - Aid has a nontrivial kernel. Suppose B isa basis for V. Discuss with your group why we can deducedet ([T - Xid] B)=0.²(c) Conversely, show that if det([T — \id]ß) = 0, then λ is an eigenvalue of T.(d) Show that det ([T – Aid]3) does not depend on our choice of basis B. That is choose adifferent basis A and show that det ([T - Aid] B) = det([T - Xid]A).(e) Discuss with your group why part (d) shows that the characteristic polynomial on thetop of this page is well defined. (f) Let A and B be similar matrices and let r € F. Prove that A - rI and B – rI aresimilar.(g) Let B be a basis of V. Show that([T - Aid]B) = [T]ß − \IConclude the characteristic polynomial of T is equal to det([T]B - XI).(h) To find all the eigenvalues À of T, we find roots of det([T]ß — XI) = 0. Use the previousparts to justify why this method works. Discuss with your groups what happens if wechoose a different basis.(i) Find the characteristic polynomial and eigenvalues of the transformations in problem 1using this method.

Answered: Image /qna-images/answer/3c23e3cf-fcad-443f-9353-3a395c704789.jpg

Answered: 2. (a) Is the definition of det(T) well…

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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Given definitions:

Definition: Let V be a vector space of dimension n, and let T : V → V be a linear transformation. The determinant of T , denoted by det(T ), is defined to be det([T ]B ), where B is any basis for V .

Definition: Let V be a vector space of dimension n. The characteristic polynomial of the linear transformation T : V → V is the polynomial in the variable λ given by det([T − λid]B), where B is any basis of V .

Definition: Let V be an n dimensional vector space and let T : V → V be a linear transformation. A basis B = {b1,··· ,bn} of V is called an eigenbasis if all bi’s are eigenvectors of T.

For part (i), problem 1 is linked here: https://www.bartleby.com/questions-and-answers/for-the-following-transformations-find-an-eigenvector-using-any-methods-you-can-think-of-including-b/7bf163e6-15fc-4ba0-91ac-24333f461731