# Why is indefinite integral not a linear transformation?

## Question:

Why is indefinite integral not a linear transformation?

## Linear Transformations:

A linear transformation is used to describe the change from function to another function through algebraic operations. This can be seen in derivatives in the derivative sum rule as the the derivative of an expression written as sum is the sum of each term's derivative term in the expression.

## Answer and Explanation:

Applying an indefinite integral to an expression of a function calculates the area under the function's curve for an unspecified domain of integration. The reason why an indefinite integral is not a linear transformation is because of the constant of integration added to the anti-derivative of a given expression. As a result of applying an indefinite integral, the constant of integration indicates that an indefinite integral creates multiple functions from one function due to the unknown value of the constant of integration. For example, the indefinite integral of {eq}2x {/eq} is {eq}x^2+c {/eq}. The indefinite integral can map {eq}2x {/eq} to {eq}x^2+1 {/eq}, {eq}x^2 {/eq}, {eq}x^2+2 {/eq}, and so on. Because the indefinite integral is not a function, the indefinite integral is not a linear transformation.

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from Statistics 101: Principles of Statistics

Chapter 2 / Lesson 13