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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


For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint: Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using.

7.

Show that the given lines intersect and find the acute angle between them.

r=2j+k+(3i-k)t1andr=7i+2k+(2i-j+k)t2

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

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

(324202423)

Let each of the following represent an active transformation of the vectors in ( x ,y )plane (axes fixed, vector rotated or reflected as in Example 3, show that each matrix is orthogonal, find its determinant and find its rotation angle, or find the line of reflectionthe

C=[0-1-10]
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