To prove that $(x + y) + z = x + (y + z)$ for all $x, y, z \in \mathbb{F}^n$, where $\mathbb{F}^n$ represents the $n$-dimensional vector space over a field $\mathbb{F}$, we need to show that the vector addition operation in $\mathbb{F}^n$ is associative.

Step 1: Define the Vectors and Their Components

Let $x, y, z \in \mathbb{F}^n$ be vectors in the $n$-dimensional vector space over the field $\mathbb{F}$. Each vector can be represented as: $x = (x_1, x_2, \dots, x_n), \quad y = (y_1, y_2, \dots, y_n), \quad z = (z_1, z_2, \dots, z_n)$ where $x_i, y_i, z_i \in \mathbb{F}$ for each $i = 1, 2, \dots, n$.

Step 2: Apply Vector Addition

The vector addition of $x$ and $y$ is defined component-wise: $x + y = (x_1 + y_1, x_2 + y_2, \dots, x_n + y_n)$ Similarly, adding $z$ to the sum $x + y$ gives: $(x + y) + z = \left((x_1 + y_1) + z_1, (x_2 + y_2) + z_2, \dots, (x_n + y_n) + z_n\right)$ On the other hand, first adding $y$ and $z$ and then adding $x$ gives: $x + (y + z) = \left(x_1 + (y_1 + z_1), x_2 + (y_2 + z_2), \dots, x_n + (y_n + z_n)\right)$

Step 3: Use Associativity in the Field $\mathbb{F}$

Since $\mathbb{F}$ is a field, addition in $\mathbb{F}$ is associative. This means that for any $a, b, c \in \mathbb{F}$, we have: $(a + b) + c = a + (b + c)$ Applying this property to each component, we get: $(x_i + y_i) + z_i = x_i + (y_i + z_i) \quad \text{for each } i = 1, 2, \dots, n$

Step 4: Conclude the Proof

Since the equality holds for each component $i$, it follows that: $(x + y) + z = x + (y + z)$ for all vectors $x, y, z \in \mathbb{F}^n$.

Summary

We have shown that vector addition in the vector space $\mathbb{F}^n$ is associative, meaning that $(x + y) + z = x + (y + z)$ holds for all vectors $x, y, z \in \mathbb{F}^n$.

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Related Questions:

1. What are the properties of a vector space over a field $\mathbb{F}$?
2. How does scalar multiplication interact with vector addition in $\mathbb{F}^n$?
3. Can you prove the commutative property of vector addition in $\mathbb{F}^n$?
4. How does associativity of addition differ between vectors and scalars?
5. What are some examples of vector spaces other than $\mathbb{F}^n$?

Tip: Understanding the component-wise operations in vector spaces helps in visualizing and proving many fundamental properties.