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!