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 vectors \(x, y, z \in \mathbb{F}^n\), we need to show that vector addition is associative in the vector space \(\mathbb{F}^n\). Here's the step-by-step proof:

### Given:
- \(x = (x_1, x_2, \ldots, x_n)\)
- \(y = (y_1, y_2, \ldots, y_n)\)
- \(z = (z_1, z_2, \ldots, z_n)\)
where \(x, y, z\) are vectors in \(\mathbb{F}^n\), the vector space of \(n\)-tuples over a field \(\mathbb{F}\).

### Step 1: Vector addition in \(\mathbb{F}^n\)
The addition of two vectors in \(\mathbb{F}^n\) is defined component-wise. For vectors \(x = (x_1, x_2, \dots, x_n)\) and \(y = (y_1, y_2, \dots, y_n)\), their sum is:
\[
x + y = (x_1 + y_1, x_2 + y_2, \dots, x_n + y_n)
\]

### Step 2: Express both sides of the equation

#### Left-hand side: \((x + y) + z\)
First, compute \(x + y\):
\[
x + y = (x_1 + y_1, x_2 + y_2, \dots, x_n + y_n)
\]
Now, add \(z = (z_1, z_2, \dots, z_n)\) to this result:
\[
(x + y) + z = ((x_1 + y_1) + z_1, (x_2 + y_2) + z_2, \dots, (x_n + y_n) + z_n)
\]

#### Right-hand side: \(x + (y + z)\)
First, compute \(y + z\):
\[
y + z = (y_1 + z_1, y_2 + z_2, \dots, y_n + z_n)
\]
Now, add \(x = (x_1, x_2, \dots, x_n)\) to this result:
\[
x + (y + z) = (x_1 + (y_1 + z_1), x_2 + (y_2 + z_2), \dots, x_n + (y_n + z_n))
\]

### Step 3: Use associativity of addition in the field \(\mathbb{F}\)
Since addition in the field \(\mathbb{F}\) is associative, we know that for any elements \(a, b, c \in \mathbb{F}\):
\[
(a + b) + c = a + (b + c)
\]
Applying this to each component of the vectors, we get:
\[
(x_i + y_i) + z_i = x_i + (y_i + z_i)
\]
for all \(i = 1, 2, \dots, n\).

### Step 4: Conclusion
Since the equality \((x_i + y_i) + z_i = x_i + (y_i + z_i)\) holds for each component \(i\), we conclude that:
\[
(x + y) + z = x + (y + z)
\]
Thus, vector addition is associative in \(\mathbb{F}^n\).

Would you like further details on this proof?

### Related Questions:
1. What is the definition of a vector space over a field \(\mathbb{F}\)?
2. Why is associativity important in vector spaces?
3. Can you show the proof for the commutativity of vector addition?
4. How does this associativity property extend to infinite-dimensional vector spaces?
5. What is the role of the field \(\mathbb{F}\) in defining the properties of \(\mathbb{F}^n\)?

### Tip:
Always check if operations (like addition or multiplication) are associative, commutative, or distributive when working with abstract algebraic structures such as vector spaces!

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

This article provides a step-by-step proof showing the associativity of vector addition in F^n, with detailed explanations of vector spaces, the field structure, and the associative property. Suitable 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