To prove that \( 1 \cdot x = x \) for all \( x \) in \( \mathbb{F}^n \), where \( \mathbb{F}^n \) denotes the vector space of \( n \)-tuples over a field \( \mathbb{F} \), we proceed as follows:

### **Proof:**

1. **Vector Space Definition:**
   \( \mathbb{F}^n \) is defined as the set of all \( n \)-tuples \( x = (x_1, x_2, \ldots, x_n) \), where each \( x_i \) is an element of the field \( \mathbb{F} \). The vector space \( \mathbb{F}^n \) is equipped with two operations: vector addition and scalar multiplication.

2. **Scalar Multiplication:**
   Scalar multiplication in \( \mathbb{F}^n \) is defined as follows: for any scalar \( \alpha \in \mathbb{F} \) and any vector \( x = (x_1, x_2, \ldots, x_n) \in \mathbb{F}^n \),
   \[
   \alpha \cdot x = \alpha \cdot (x_1, x_2, \ldots, x_n) = (\alpha \cdot x_1, \alpha \cdot x_2, \ldots, \alpha \cdot x_n)
   \]
   where each \( \alpha \cdot x_i \) is the multiplication of the scalar \( \alpha \) with the component \( x_i \) in the field \( \mathbb{F} \).

3. **Multiplicative Identity:**
   The field \( \mathbb{F} \) contains a multiplicative identity element, denoted by 1, such that for any \( y \in \mathbb{F} \),
   \[
   1 \cdot y = y
   \]
   This property applies to each component of the vector \( x \).

4. **Applying Scalar Multiplication with 1:**
   Let \( x = (x_1, x_2, \ldots, x_n) \) be an arbitrary vector in \( \mathbb{F}^n \). Then, multiplying this vector by the scalar 1 gives:
   \[
   1 \cdot x = 1 \cdot (x_1, x_2, \ldots, x_n) = (1 \cdot x_1, 1 \cdot x_2, \ldots, 1 \cdot x_n)
   \]
   Using the property of the multiplicative identity in the field \( \mathbb{F} \),
   \[
   1 \cdot x_i = x_i \quad \text{for each } i = 1, 2, \ldots, n
   \]
   Therefore,
   \[
   1 \cdot x = (x_1, x_2, \ldots, x_n) = x
   \]

5. **Conclusion:**
   We have shown that for any vector \( x \) in the vector space \( \mathbb{F}^n \), multiplying by the scalar 1 leaves the vector unchanged, i.e., \( 1 \cdot x = x \). This completes the proof.

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Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What is the difference between a field \( \mathbb{F} \) and a vector space \( \mathbb{F}^n \)?
2. How does scalar multiplication in \( \mathbb{F}^n \) relate to matrix multiplication?
3. Can you prove that \( 0 \cdot x = 0 \) for all \( x \in \mathbb{F}^n \)?
4. What role does the distributive property play in vector spaces?
5. How can the concept of basis vectors be used to understand operations in \( \mathbb{F}^n \)?

**Tip:** Understanding the interaction between scalars and vectors is crucial when working with vector spaces, especially when dealing with transformations and linear mappings.

Learn how to prove that 1 times x equals x for all vectors x in the vector space Fn. This proof involves concepts from vector spaces, scalar multiplication, and field theory. Suitable for advanced undergraduate students studying algebra and linear algebra.

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