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.