To show that \( 1 \cdot x = x \) for all \( x \) in \( \mathbb{F}^N \) (where \( \mathbb{F} \) is a field and \( \mathbb{F}^N \) represents the \( N \)-dimensional vector space over \( \mathbb{F} \)), we can proceed as follows:

### 1. Understanding the Problem
In this context, \( 1 \) represents the multiplicative identity in the field \( \mathbb{F} \), and \( x \) is a vector in \( \mathbb{F}^N \). The goal is to show that multiplying any vector \( x \) by 1 results in the same vector \( x \).

### 2. Vector Representation
Let \( x \) be a vector in \( \mathbb{F}^N \). We can write \( x \) as:
\[
x = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_N \end{pmatrix}
\]
where each \( x_i \in \mathbb{F} \) for \( i = 1, 2, \ldots, N \).

### 3. Scalar Multiplication
Scalar multiplication in a vector space is defined component-wise. If we multiply the vector \( x \) by the scalar \( 1 \) (the multiplicative identity in \( \mathbb{F} \)), we get:
\[
1 \cdot x = 1 \cdot \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_N \end{pmatrix} = \begin{pmatrix} 1 \cdot x_1 \\ 1 \cdot x_2 \\ \vdots \\ 1 \cdot x_N \end{pmatrix}
\]

### 4. Property of the Multiplicative Identity
In any field \( \mathbb{F} \), the multiplicative identity \( 1 \) has the property that for any element \( x_i \in \mathbb{F} \), \( 1 \cdot x_i = x_i \). Applying this to each component:
\[
1 \cdot x_i = x_i \quad \text{for each } i = 1, 2, \ldots, N
\]

### 5. Conclusion
Therefore, we have:
\[
1 \cdot x = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_N \end{pmatrix} = x
\]
This shows that \( 1 \cdot x = x \) for any vector \( x \) in \( \mathbb{F}^N \).

### Summary
The identity \( 1 \cdot x = x \) holds for all vectors \( x \) in \( \mathbb{F}^N \) because scalar multiplication is defined component-wise and the multiplicative identity \( 1 \) in a field satisfies \( 1 \cdot x_i = x_i \) for each component.

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the significance of the field \( \mathbb{F} \) in this proof?
2. How does scalar multiplication differ in different fields, such as \( \mathbb{R}^N \) and \( \mathbb{C}^N \)?
3. What are some examples of vector spaces that are not over fields?
4. How does this identity help in simplifying vector equations?
5. Can this property be generalized to other algebraic structures?

**Tip:** Always remember that the identity element in any operation is defined as the element that leaves other elements unchanged when combined with them under that operation.

Learn how to prove the identity 1x = x holds for all vectors x in the N-dimensional vector space over a field F. This problem explores vector spaces, scalar multiplication, and the properties of fields, showcasing the fundamental concept of the multiplicative identity.

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