prove that the sum of two linear operators is a linear operator 

To prove that the sum of two linear operators is itself a linear operator, let's begin by reviewing the definition of a linear operator.

### Definition of a Linear Operator:
A linear operator \( T \) on a vector space \( V \) satisfies the following two properties for all vectors \( \mathbf{u}, \mathbf{v} \in V \) and any scalar \( c \):

1. **Additivity**: \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
2. **Homogeneity (Scalar Multiplication)**: \( T(c\mathbf{u}) = cT(\mathbf{u}) \)

### Given:
Let \( T_1 \) and \( T_2 \) be linear operators on a vector space \( V \).

### To Prove:
The sum \( T_1 + T_2 \) is a linear operator on \( V \).

### Proof:

1. **Additivity**: We need to show that \( (T_1 + T_2)(\mathbf{u} + \mathbf{v}) = (T_1 + T_2)(\mathbf{u}) + (T_1 + T_2)(\mathbf{v}) \) for all \( \mathbf{u}, \mathbf{v} \in V \).

   \[
   \begin{aligned}
   (T_1 + T_2)(\mathbf{u} + \mathbf{v}) &= T_1(\mathbf{u} + \mathbf{v}) + T_2(\mathbf{u} + \mathbf{v}) & \text{(by definition of operator addition)} \\
   &= \big(T_1(\mathbf{u}) + T_1(\mathbf{v})\big) + \big(T_2(\mathbf{u}) + T_2(\mathbf{v})\big) & \text{(since \( T_1 \) and \( T_2 \) are linear)} \\
   &= \big(T_1(\mathbf{u}) + T_2(\mathbf{u})\big) + \big(T_1(\mathbf{v}) + T_2(\mathbf{v})\big) & \text{(rearranging terms)} \\
   &= (T_1 + T_2)(\mathbf{u}) + (T_1 + T_2)(\mathbf{v}) & \text{(by definition of operator addition)}
   \end{aligned}
   \]
   
   Therefore, the additivity property holds for \( T_1 + T_2 \).

2. **Homogeneity (Scalar Multiplication)**: We need to show that \( (T_1 + T_2)(c\mathbf{u}) = c(T_1 + T_2)(\mathbf{u}) \) for any scalar \( c \) and vector \( \mathbf{u} \in V \).

   \[
   \begin{aligned}
   (T_1 + T_2)(c\mathbf{u}) &= T_1(c\mathbf{u}) + T_2(c\mathbf{u}) & \text{(by definition of operator addition)} \\
   &= cT_1(\mathbf{u}) + cT_2(\mathbf{u}) & \text{(since \( T_1 \) and \( T_2 \) are linear)} \\
   &= c\big(T_1(\mathbf{u}) + T_2(\mathbf{u})\big) & \text{(factoring out the scalar \( c \))} \\
   &= c(T_1 + T_2)(\mathbf{u}) & \text{(by definition of operator addition)}
   \end{aligned}
   \]
   
   Therefore, the homogeneity property holds for \( T_1 + T_2 \).

### Conclusion:
Since both additivity and homogeneity properties hold, we have shown that the sum \( T_1 + T_2 \) is indeed a linear operator on the vector space \( V \).

Would you like more details or have any questions? Here are five related questions to expand on this topic:

1. What are some examples of linear operators in different vector spaces?
2. How does the composition of linear operators relate to linearity?
3. Can the sum of a linear and a nonlinear operator be a linear operator?
4. How can you prove that the difference of two linear operators is also a linear operator?
5. What happens to linearity when operators are defined on infinite-dimensional vector spaces?

**Tip:** When working with linear operators, it's often helpful to test both additivity and homogeneity with specific examples to gain intuition before proving the general case.

Prove that the sum of two linear operators is a linear operator.

Explore the detailed proof that demonstrates how the sum of two linear operators remains a linear operator. This explanation covers key concepts in linear algebra, including additivity and homogeneity, making it suitable for undergraduate students.

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