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The goal of this exercise is to show that:

\[
(a + b)x = ax + bx
\]

for all \( a, b \in \mathbb{F} \) and for all \( x \in \mathbb{F}^n \).

This is an expression that demonstrates the distributive property of scalar multiplication over vector addition in a vector space.

---

### **Proof**

We will prove this step-by-step, assuming \( a, b \) are scalars in a field \( \mathbb{F} \) and \( x \) is a vector in \( \mathbb{F}^n \).

1. **Consider \( x \) as a vector in \( \mathbb{F}^n \)**:  
   Let \( x = (x_1, x_2, \ldots, x_n) \) where each \( x_i \in \mathbb{F} \).

2. **Define scalar multiplication of \( (a + b) \) with \( x \)**:  
   By the definition of scalar multiplication in \( \mathbb{F}^n \), we have:  
   \[
   (a + b)x = ((a + b)x_1, (a + b)x_2, \ldots, (a + b)x_n).
   \]

3. **Distribute \( (a + b) \) in each component**:  
   Using the distributive property of the field \( \mathbb{F} \), we can write:  
   \[
   (a + b)x_i = ax_i + bx_i \quad \text{for each } i = 1, 2, \ldots, n.
   \]

   Therefore:  
   \[
   (a + b)x = (ax_1 + bx_1, ax_2 + bx_2, \ldots, ax_n + bx_n).
   \]

4. **Rewrite the vector sum**:  
   Now, observe that the right-hand side can be expressed as the sum of two vectors:  
   \[
   (ax_1 + bx_1, ax_2 + bx_2, \ldots, ax_n + bx_n) = (ax_1, ax_2, \ldots, ax_n) + (bx_1, bx_2, \ldots, bx_n).
   \]

5. **Express in terms of scalar multiplication**:  
   This is the same as:  
   \[
   ax + bx.
   \]

Thus, we have shown that:

\[
(a + b)x = ax + bx.
\]

---

### Conclusion

This proof verifies the distributive property of scalar multiplication over vector addition in a vector space.

---

### Related Questions

1. What is the definition of scalar multiplication in a vector space?
2. How can you prove the distributive property in general for fields other than \( \mathbb{R} \)?
3. How does the distributive property extend to more than two scalars?
4. What are some examples of vector spaces where this property holds?
5. What role does the field \( \mathbb{F} \) play in defining vector spaces?

**Tip:** The distributive property is a fundamental concept in linear algebra and ensures the compatibility of scalar multiplication with vector addition.

Show that (a + b)x = ax + bx for all a, b ∈ F and all x ∈ F^n.

Understand and prove the distributive property of scalar multiplication over vector addition: (a + b)x = ax + bx for all scalars a, b in a field F and all vectors x in F^n. Explore key linear algebra concepts with a step-by-step solution.

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