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Chapter 7: Q50E (page 358)

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.

T(f(x))=f(x-3)fromP2 toP2 .Is Tdiagonalizable?

Short Answer

Expert verified

T is not diagonalizable for T(f(x))=f(x-3)fromP2to P2, having the eigenvalues as:

位1=1位2=1位3=1

The eigenspaces are:

E1=span(1)

Step by step solution

01

Define the criteria for T to be diagonalizable and eigenvectors

T is diagonalizable if and only if the vectors in the basis are all eigenvectors of T.

An eigenvector is a non-zero vector that changes mostly by a scale factor during linear transformation.

02

Find the linear transformation with given condition

The linear transformation forf(x)=ax2+bx+c is given by:

T(f(x))=Tax2+bx+c=a(x鈭�3)2+b(x鈭�3)+c=ax2鈭�6ax+9a+bx鈭�3b+c=ax2+(鈭�6a+b)x+(9a鈭�3b+c)

The basis for the transformation is 尾=x2,x,1 .

The matrix of the transformation will be:

B=100-6109-31

03

Find the eigenvalues and eigenspaces

The eigenvalues of T are:

位1=1位2=1位3=1

The eigenspaces are:

E1=span(1)

This means that T is not diagonalizable.

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Most popular questions from this chapter

suppose a certain4脳4 matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.

Find an eigenbasis of given matrix and diagonalize it.

A=(1326)

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[-1-1-1-1-1-1-1-1-1]

(a) Give an example of a 3 脳 3 matrix A with as many nonzero entries as possible such that both span(e鈫�1) and span(e鈫�1,e鈫�2) are A-invariant subspaces of . See Exercise 65.

(b) Consider the linear space Vof all 3 脳 3 matrices A such that both span (e鈫�1) and span (e鈫�1,e鈫�2) are A-invariant subspaces of R3 . Describe the space V (the matrices in V 鈥渉ave a name鈥�), and determine the dimension of V.

Arguing geometrically, find all eigen vectors and eigen values of the linear transformations. In each case, find an eigen basis if you can, and thus determine whether the given transformation is diagonalizable.

The linear transformation withT(v鈫�)=v鈫�, and T(w鈫�)=v鈫�+w鈫�for the vectorsv鈫�andw鈫�inR2 sketched below.
See all solutions

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