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

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 \(x \in F^n\).

---

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.

This detailed solution shows how to prove the distributive property of scalar multiplication in vector spaces: (a+b)x = ax + bx for all a, b in field F and all vectors x in F^n. Learn key concepts such as scalar multiplication, vector addition, and the distributive property. Ideal for undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation