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Chapter 7: Q48E (page 358)

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

T(f(x))=f(2x)from P2 to P2.Is Tdiagonalizable?

Short Answer

Expert verified

T is diagonalizable for T(f(x))=f(2x)fromP2 to P2, having the eigenvalues as:

位1=1位2=2位3=4

The eigenspaces are:

EE=san(1}E2=lin{x)E4=spanx2

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尾 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))=T(ax2+bx+c)=4ax2+2bx+c

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

The matrix of the transformation will be:

B=100020004

03

Find the eigenvalues and eigenspaces

The eigenvalues of T are:

位=1位2=2位3=4

The eigenspaces are:

E1=span(1)E2=span{x}E4=spanx2

This means that T is diagonalizable.

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Most popular questions from this chapter

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