Show that (𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝐅^𝑛 .

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 \(x, y, z \in \mathbb{F}^n\).

Do you want more details or have any questions?

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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.

Learn how to prove that (x + y) + z = x + (y + z) holds for all vectors x, y, z in the n-dimensional vector space over a field. This proof involves understanding vector addition and the associativity property in field theory.

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