Math Problem Statement

Show that 1x = x for all in F^n.

Solution

To prove that 1x=x1 \cdot x = x for all xx in Fn\mathbb{F}^n, where Fn\mathbb{F}^n denotes the vector space of nn-tuples over a field F\mathbb{F}, we proceed as follows:

Proof:

  1. Vector Space Definition: Fn\mathbb{F}^n is defined as the set of all nn-tuples x=(x1,x2,,xn)x = (x_1, x_2, \ldots, x_n), where each xix_i is an element of the field F\mathbb{F}. The vector space Fn\mathbb{F}^n is equipped with two operations: vector addition and scalar multiplication.

  2. Scalar Multiplication: Scalar multiplication in Fn\mathbb{F}^n is defined as follows: for any scalar αF\alpha \in \mathbb{F} and any vector x=(x1,x2,,xn)Fnx = (x_1, x_2, \ldots, x_n) \in \mathbb{F}^n, αx=α(x1,x2,,xn)=(αx1,αx2,,αxn)\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 αxi\alpha \cdot x_i is the multiplication of the scalar α\alpha with the component xix_i in the field F\mathbb{F}.

  3. Multiplicative Identity: The field F\mathbb{F} contains a multiplicative identity element, denoted by 1, such that for any yFy \in \mathbb{F}, 1y=y1 \cdot y = y This property applies to each component of the vector xx.

  4. Applying Scalar Multiplication with 1: Let x=(x1,x2,,xn)x = (x_1, x_2, \ldots, x_n) be an arbitrary vector in Fn\mathbb{F}^n. Then, multiplying this vector by the scalar 1 gives: 1x=1(x1,x2,,xn)=(1x1,1x2,,1xn)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 F\mathbb{F}, 1xi=xifor each i=1,2,,n1 \cdot x_i = x_i \quad \text{for each } i = 1, 2, \ldots, n Therefore, 1x=(x1,x2,,xn)=x1 \cdot x = (x_1, x_2, \ldots, x_n) = x

  5. Conclusion: We have shown that for any vector xx in the vector space Fn\mathbb{F}^n, multiplying by the scalar 1 leaves the vector unchanged, i.e., 1x=x1 \cdot x = x. This completes the proof.


Would you like more details or have any questions?

Here are 5 related questions to consider:

  1. What is the difference between a field F\mathbb{F} and a vector space Fn\mathbb{F}^n?
  2. How does scalar multiplication in Fn\mathbb{F}^n relate to matrix multiplication?
  3. Can you prove that 0x=00 \cdot x = 0 for all xFnx \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 Fn\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.

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Math Problem Analysis

Mathematical Concepts

Vector Spaces
Scalar Multiplication
Field Theory

Formulas

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Theorems

Multiplicative Identity

Suitable Grade Level

Advanced Undergraduate