Question

Question:Are the following operators linear?

7. Definite integral with respect to x from 0 to 1; the objects being operated on are functions of x.

Short answer

The definite integral with respect to x from 0 to 1 with the objects being operated on are functions of x representing a linear operator.

Step-by-step solution

Definition of Linear Operator

The term"linear operator" is very frequently used in linear algebra to refer to linear maps (i.e., functions that define vector addition and scalar multiplication) that have the additional feature of mapping a vector space into itself.

Given parameters

A Definite integral operator is given in this problem.

Check to see whether the given definite integral operator is linear or not.

Find out if the given definite integral operator is linear

Shown that the given definite integral operator operates on the function f(x) by integrating it with respect to 0 to 1 with the help of linear operator definition.

An operator O represents a linear operator if the given relations are satisfied which are defined as $O(A+B)=O\left(A\right)+O\left(B\right)$and $O\left(kA\right)=kO\left(A\right)$, where k is a number and A and B are numbers, functions, vectors, etc.

For the definite integral $I=\underset{a=0}{\overset{b=1}{\int }}f\left(x\right)dx$, from calculus, $\underset{a=0}{\overset{b=1}{\int }}\left[f\left(x\right)+g\left(x\right)\right]dx=\underset{a=0}{\overset{b=1}{\int }}f\left(x\right)dx+\underset{a=0}{\overset{b=1}{\int }}g\left(x\right)dx$, and $\underset{a=0}{\overset{b=1}{\int }}kf\left(x\right)dx=k\underset{a=0}{\overset{b=1}{\int }}f\left(x\right)dx$, where k is a constant.

Therefore, the given definite integral with respect to x from 0 to 1 with the objects being operated on are functions of representing a linear operator.