Proofs in Linear Algebra: Countable Sets, Algebraic Numbers, and Fields

In summary, the conversation discusses various proofs and representations related to linear algebra and fields. The first topic involves proving that any linear well-ordered set on a countable set C is induced from the canonical order on natural numbers via a bijection. The second topic is about proving that all algebraic numbers form a countable subfield of the complex numbers. The third topic discusses finding a representation of the ring C[[x]] as an inverse limit of rings with finite dimension over C. Finally, the fourth topic is a straightforward proof that the field Qp is uncountable, involving a surjection onto the non-negative real numbers using a representation of Qp.
  • #1
blaster
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Linear Algebra proof

I would appreciate any help with any of the foolowing:

1. Let C be a countable set. Prove that any linear well-ordered on C with the property that whatever c in C there are only finitely elements c` with c`<c, is unduced from the canonical order on N via a bijection N-> C. (N - natural no)

2. Prove that all algebraic numbers (all roots of polynomials with rational coefficients) form a countable subfield of C (complex).

3. Find a representation of the ring C[[x]] as an inverse limit of rings which have finite dimension over C.(complex).

4. Prove that the field Qp is uncountable. (Qp=field of fractions of Zp)
 
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  • #2
Number 4 is straight forward. You just need a surjection onto the reals (well in this case the non-negative reals which are also uncountable).

Hint. Given a number p any real number can be written in the form [tex]\sum_{j=n}^\infty a_j p^{-j} [/tex] with [tex]0 \le a_j < p [/tex] for some [tex]n\in \mathbb{Z}[/tex]. Do you know any representations of Qp that is similar? The surjection doesn't not need any homomorphic properties.

Easier than a diagonal argument. Though there is also a diagonal argument.
 
  • #3



1. To prove that any linear well-ordered on C with the given property is induced from the canonical order on N via a bijection, we need to show that there exists a bijection f: N -> C such that for any two elements n, m in N, n < m if and only if f(n) < f(m) in C.

First, we know that C is countable, which means that there exists a bijection g: N -> C. Let's define a new function h: N -> C such that h(n) = g(n+1). This function essentially shifts the elements of C by one, so that the first element of C is now mapped to the second element, the second element to the third, and so on.

Next, let's define a new function f: N -> C such that f(n) = g(1) if n = 0, and f(n) = h(n-1) if n > 0. This function essentially maps the first element of N to the first element of C, and then maps the remaining elements of N to the remaining elements of C in the same order.

Now, let's consider any two elements n, m in N. If n < m, then we know that f(n) = h(n-1) < h(m-1) = f(m) in C, since h is a bijection. And if f(n) < f(m) in C, then we can easily see that n < m, since f(n) = g(1) if n = 0, and f(n) = h(n-1) if n > 0.

Therefore, we have shown that there exists a bijection f: N -> C such that for any two elements n, m in N, n < m if and only if f(n) < f(m) in C. This proves that the linear well-order on C is induced from the canonical order on N via a bijection, as desired.

2. To prove that all algebraic numbers form a countable subfield of C, we first need to show that the set of all polynomials with rational coefficients is countable. This is because each polynomial can be represented as a finite sequence of rational coefficients, and the set of all finite sequences of rational numbers is countable.

Next, we know that the set of all roots of a polynomial with rational coefficients is finite. This
 

FAQ: Proofs in Linear Algebra: Countable Sets, Algebraic Numbers, and Fields

What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with linear equations and their representation in vector spaces. It involves the study of linear transformations and their properties, as well as systems of linear equations and their solutions.

Why are proofs important in Linear Algebra?

Proofs are important in Linear Algebra because they provide a logical and rigorous way to demonstrate the validity of mathematical statements and theorems. They allow us to understand and justify the concepts and techniques used in Linear Algebra, and to apply them confidently in various problem-solving situations.

What are some common techniques used in Linear Algebra proofs?

Some common techniques used in Linear Algebra proofs include mathematical induction, direct proof, proof by contradiction, and proof by contrapositive. Other techniques may involve using properties of matrices, vector spaces, and linear transformations, as well as techniques from other branches of mathematics such as calculus and geometry.

How can I improve my skills in writing Linear Algebra proofs?

To improve your skills in writing Linear Algebra proofs, it is important to have a strong understanding of the fundamental concepts and theorems in Linear Algebra. Practice is also crucial, so solving a variety of problems and attempting different proof techniques can help strengthen your skills. Additionally, seeking guidance from a mentor or tutor can also be beneficial.

Can Linear Algebra proofs be applied in real-world situations?

Yes, Linear Algebra proofs have many real-world applications. For example, they are used in computer graphics, data analysis, and engineering to model and solve problems involving linear systems and transformations. Linear Algebra proofs also play a significant role in machine learning and artificial intelligence algorithms.

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