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Chapter 3: Q9P (page 130)

Question: Find the square; operate on numbers or on functions.

Short Answer

Expert verified

The square of any function which operates on the function f(x) by squaring it does not represent a linear operator.

Step by step solution

01

Definition of the linear operator:

An operator is said to be a linear operator if the following relations are satisfied ifO(A+B)=O(A)+O(B)andO(kA)=kO(A). .

Where, K is a number, and A and, B are numbers, functions, vectors, etc.

02

Operation on square function for O(A+B)=O(A)+O(B) :

The square function which operates on the function is given by

Q(x)=[f(x)]2

It is known that

fx+gx2=f2x+2fxgx+g2x=fx2+gx2+2fxgx+g2x=fx2+gx2

03

Operation on square function for  O(kA)=kO(A) :

Evaluate the square of (kf(x))2.

(kf(x))2=k2f2x=k2fx2≠k2fx2

Therefore, it is shown that the square of any function which operate on the function fx by squaring it does not represent a linear operator.

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Most popular questions from this chapter

Verify the details in the discussion of the matrices in (11.31).

Find the Eigen values and Eigen vectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer.

(230320001)

Let each of the following matrices Mdescribe a deformation of the (x,y)plane For each given M find: the Eigen values and eigenvectors of the transformation, the matrix Cwhich DiagonalizesM and specifies the rotation to new axes(x',y')along the eigenvectors, and the matrix D which gives the deformation relative to the new axes. Describe the deformation relative to the new axes.

(3449)

Write the matrices which produce a rotation about the axis, or that rotation combined with a reflection through the (y,z) plane.

Verify the calculations in (6.15) ,(6.16), and (6.17) .
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