Are the following operators linear? Definite integral with respect to \(x\) from 0 to \(1 ;\) the objects being operated on are functions of \(x\).

Understanding the concept of additivity is crucial when discussing linear operators. Additivity means that the sum of the operator applied to two functions is equal to the operator applied to the sum of the two functions. For example, when dealing with the definite integral operator from 0 to 1, if we have two functions, say, \( f(x) \) and \( g(x) \), additivity states that:
\[ T(f(x) + g(x)) = \ \ =\int_{0}^{1} [f(x) + g(x)] \, dx \]
Using the property of integrals:
\[ \ = \int_{0}^{1} f(x) \, dx + \int_{0}^{1} g(x) \, dx \]
This shows that:
\[ T(f(x) + g(x)) = T(f(x)) + T(g(x)) \]
This property is essential because it helps define the behavior of linear operators. For any linear operator, additivity must hold true.

Next, let's dive into homogeneity, another vital property of linear operators. Homogeneity suggests that when we scale a function by a constant and then apply the operator, it should be equivalent to scaling the result of the operator by the same constant. More concretely, for a function \( f(x) \) and a scalar \( c \), considering our definite integral operator from 0 to 1:
\[ T(c \times f(x)) = \ \ =\int_{0}^{1} c \times f(x) \, dx \]
Applying the properties of integrals again:
\[ \ = c \times \int_{0}^{1} f(x) \, dx \]
This simplifies to:
\[ T(c \times f(x)) = c \times T(f(x)) \]
This homogeneity property proves essential as it keeps the scalability of the functions consistent through the linear operator.

To understand linear operators better, we need to grasp the concept of definite integrals. A definite integral represents the area under a curve within certain bounds. For instance, the definite integral of a function \( f(x) \) from 0 to 1 is given by:
\[ \int_{0}^{1} f(x) \, dx \]
Think of this as summing up all the tiny areas under the function \( f(x) \) between the points 0 and 1 on the x-axis. This integral calculation is not just an abstract concept; it applies to real-world scenarios like finding the total distance traveled given a speed function. When considering it as an operator, the definite integral has properties that align well with the principles of linearity, such as additivity and homogeneity.

Finally, let's explore some basic integration properties which are foundational for understanding the linearity of integral operators.
1. **Linearity Property:** This is made up of additivity and homogeneity, previously covered. It states that the integral of a sum is the sum of the integrals, and the integral of a scaled function is the scale times the integral of the function.
2. **Bounded Intervals:** Integrals are typically computed over a specific interval, ensuring finite results.
3. **Fundamental Theorem of Calculus:** This links the concept of definite integrals with antiderivatives, providing a comprehensive framework for understanding how integrals work.
These properties solidify the behavior of linear operators, reinforcing why the definite integral from 0 to 1 behaves linearly.