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Chapter 3: Q10P (page 130)

Find the reciprocal; operate on numbers or on functions.

Short Answer

Expert verified

The reciprocal operator of any function which operates on the function f(x)by inverting 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 if

O(A+B)=O(A)+O(B)andO(kA)=kO(A)

whereis a number, andand,are numbers, functions, vectors, etc.

02

Operation on reciprocal function for  O(A+B)=O(A)+O(B) 

Find the Reciprocal function which operates on the function f(x)..

Q(x)=1f(x)

Evaluate role="math" localid="1658990953893" r(f(x)+g(x)).

r(f(x)+g(x))=1(f(x)+g(x))≠1f(x)+1g(x)≠rfx+rfx

03

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

Evaluater(kf(x)). .

r(kf(x))=1kfx=1k1fx=1krfx=k1fx

Therefore, it is shown that the reciprocal operator of any function which operates on the function fxby inverting it does not represent a linear operator.

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Most popular questions from this chapter

Let A=2i^-j^+2k^ . (a) Find a unit vector in the same direction as A . Hint: Divide A by |A|. (b) Find a vector in the same direction as A but of magnitude 12 . (c) Find a vector perpendicular to A . Hint: There are many such vectors; you are to find one of them. (d) Find a unit vector perpendicular to A . See hint in (a).

In Problems8to15,use(8.5)to show that the given functions are linearly independent.

13.sin⁡x,sin⁡2x

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.

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solve the set of equations by the method of finding the inverse of the coefficient matrix.

{2x+3y=-15x+4y=8

(a) Prove that.(AB)t=BtAt Hint: See(9.10).

(b) Verify (9.11), that is, show that (9.10) applies to a product of any number of matrices. Hint: Use (9.10)and (9.8).
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