To show that $(a + b)x = ax + bx$ for all $a, b \in F$ and $x \in F^n$, we will proceed by using basic properties of vector spaces and scalar multiplication.

Given:

• $a, b$ are scalars in a field $F$.
• $x \in F^n$, meaning $x$ is an element of the $n$-dimensional vector space over the field $F$.
• We want to prove that scalar multiplication distributes over vector addition: $(a + b)x = ax + bx$.

Proof:

1. Let $x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$ where each $x_i \in F$.

2. By the definition of scalar multiplication in a vector space, for any scalar $c \in F$ and vector $x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, we have: $cx = \begin{bmatrix} c x_1 \\ c x_2 \\ \vdots \\ c x_n \end{bmatrix}.$

3. Now, consider the left-hand side of the equation $(a + b)x$: $(a + b)x = \begin{bmatrix} (a + b)x_1 \\ (a + b)x_2 \\ \vdots \\ (a + b)x_n \end{bmatrix}.$

4. Using the distributive property of the field $F$ (since $F$ is a field, scalar multiplication distributes over addition in $F$), we can distribute $a + b$ over each $x_i$: $(a + b)x_i = ax_i + bx_i \quad \text{for each} \ i.$ Therefore: $(a + b)x = \begin{bmatrix} ax_1 + bx_1 \\ ax_2 + bx_2 \\ \vdots \\ ax_n + bx_n \end{bmatrix}.$

5. Now consider the right-hand side $ax + bx$: $ax = \begin{bmatrix} ax_1 \\ ax_2 \\ \vdots \\ ax_n \end{bmatrix}, \quad bx = \begin{bmatrix} bx_1 \\ bx_2 \\ \vdots \\ bx_n \end{bmatrix}.$ Adding these two vectors component-wise gives: $ax + bx = \begin{bmatrix} ax_1 + bx_1 \\ ax_2 + bx_2 \\ \vdots \\ ax_n + bx_n \end{bmatrix}.$

6. Since both the left-hand side $(a + b)x$ and the right-hand side $ax + bx$ yield the same vector, we conclude that: $(a + b)x = ax + bx.$

Thus, we have proven that $(a + b)x = ax + bx$ for all scalars $a, b \in F$ and all vectors $x \in F^n$.

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Would you like more details on any step or have further questions?

Related Questions:

1. What are the properties of vector spaces that allow such proofs?
2. Can this result be generalized to infinite-dimensional vector spaces?
3. How does scalar multiplication distribute over vector addition in non-commutative fields?
4. What is the significance of scalar multiplication in linear transformations?
5. How can this property be used to prove other linear algebra properties?

Tip:

Understanding the distributive property in vector spaces is crucial for working with linear transformations and matrix operations.