Let T : R² → R² be the linear map T(x, y)=(y,x).(a) What is the standard matrix of T?(b) Give a geometric description of T using the word reflection.(c) Find the eigenvalues of T (just using the definition).(d) Find an eigenvector for each eigenvalue.(e) For each of your eigenvectors v from (d), graph v and Tv = \v.

Answered: Let T: R² R² be the linear map T(x, y)…

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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Question

Let T : R² → R² be the linear map T(x, y)
=
(y,x).
(a) What is the standard matrix of T?
(b) Give a geometric description of T using the word reflection.
(c) Find the eigenvalues of T (just using the definition).
(d) Find an eigenvector for each eigenvalue.
(e) For each of your eigenvectors v from (d), graph v and Tv = \v.

Expert Solution

Step 1

Note : " As per our guidelines we will solve three subparts . If you want any specific subpart to be solved please specify that subpart or post only that subpart . "

(.) Linear map $T : ℝ^{2} \to ℝ^{2}$ given by ,

$T (x, y) = (y, x)$

(.) Standard basis of $ℝ^{2}$ is $\{(1, 0), (0, 1)\}$ .

(.) Let $T : V (F) \to V (F)$ be a linear transformation , then $λ \in F$ is a eigen value of $T$ if and only if $T - λ I$ is singular . Here $I$ is an identity transformation and $V$ is a vector space over the field $F$ .

(.) Singular linear transformation : A linear transformation $T$ is said to be singular if there exists a vector $u \neq 0$ such that $T (u) = 0$ .