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Chapter 7: Q51E (page 358)

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

T(f)=fffrom P to P

Short Answer

Expert verified

The eigenvector is 位=0.

The eigenspace is E0=span(1) .

Step by step solution

01

Define eigenvectors

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

02

Find the eigenvector and eigenspace for the function

The polynomials defined by f having derivatives as scalar are constant functions given by f(x)=c鈭�R.

So, there is only one eigenvalue of T that is 位=0 .

The eigenspace is E0=span(1).

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Most popular questions from this chapter

Give an example of a matrixAof rank 1 that fails to be diagonalizable.

(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.

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.

[04-14]

Consider ann脳n matrix such that the sum of the entries in each row is . Show that the vector

e鈫�=11..1

InRn is an eigenvector of A. What is the corresponding eigenvalue?

7: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.

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