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Chapter 7: Q52E (page 358)

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

T(f(x))=x(f'(x))ffrom P to P

Short Answer

Expert verified

The eigenvector is 位n=n .

The eigenspace is En=span(xn),鈭赌n鈭�N .

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鈭�P,

f(x)=anxn+an-1xn-1+鈰�+a2x2+a1x+a0

So, the transformation will be:

T(f(x))=x脳nanxn鈭�1+(n鈭�1)an鈭�1xn鈭�2+鈥�+2a2x+a1=nanxn+(n鈭�1)an鈭�1xn鈭�1+鈥�+2a2x2+a1x

So, the eigenvalues of T are 位n=n.

The eigenspace is defined as En=span(xn),鈭赌n鈭�N.

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