# Mathematical physics - writing proofs

• Hypercube
In summary, the conversation is about self-studying mathematical physics and writing proofs. The question is to prove that all endomorphisms on ℂ produce a constant-multiple of the input vector. The attempt at a solution involves analyzing and reiterating the question in one's own words and then attempting the proof by considering different cases. It is mentioned that further thought is needed to address potential issues with the proof.
Hypercube
Hi there!

So for about a month, I've been self-studying mathematical physics using Mathematical Physics by Hassani. It's a big change for me, after only Stewart calculus, Boyce DE and some linear algebra course, but it's been loads of fun! Now, writing proofs is something fairly new to me, and I think it would be good if someone more experienced goes through and confirms whether what I write is correct, or at the very least good enough.

1. Homework Statement

## Homework Equations

L(ℂ, ℂ) refers to the set of endomorphisms on ℂ.

## The Attempt at a Solution

Step 1. Analyse and reiterate the question in your own words.

Prove that all endomorphisms on ℂ produce constant-multiple of the input vector.Step 2. Attempt the proof.

Let T be an endomorphism on ℂ such that T(a) = b, and assume that b ≠ αa. Since the range of T is a subspace of ℂ, b must also be in ℂ. This leads to contradiction because ℂ now has a number of linearly independent vectors that exceeds its dimension (1). Hence, ba.===================Is this proof any good? I have spent quite a bit of time on it, and it is the best I could come up with.

Thanks

Hypercube said:
Hi there!

So for about a month, I've been self-studying mathematical physics using Mathematical Physics by Hassani. It's a big change for me, after only Stewart calculus, Boyce DE and some linear algebra course, but it's been loads of fun! Now, writing proofs is something fairly new to me, and I think it would be good if someone more experienced goes through and confirms whether what I write is correct, or at the very least good enough.

1. Homework Statement

View attachment 110742

## Homework Equations

L(ℂ, ℂ) refers to the set of endomorphisms on ℂ.

## The Attempt at a Solution

Step 1. Analyse and reiterate the question in your own words.

Prove that all endomorphisms on ℂ produce constant-multiple of the input vector.Step 2. Attempt the proof.

Let T be an endomorphism on ℂ such that T(a) = b, and assume that b ≠ αa. Since the range of T is a subspace of ℂ, b must also be in ℂ. This leads to contradiction because ℂ now has a number of linearly independent vectors that exceeds its dimension (1). Hence, ba.===================Is this proof any good? I have spent quite a bit of time on it, and it is the best I could come up with.

Thanks
Looks good. I only would spent some thoughts on the cases: Why is ##\alpha_a## the same for all ##a##? You only proved it for a single one. They could all be different. And it could be, that ##b=0##, then linear dependence is automatically true and the dimension argument breaks down. Could there as well exist ##b\neq 0## (for a different ##a##)?

Hypercube
This never even occurred to me. Indeed, what if a belongs to ker(T)? I will spend some more time on this and see if I can modify the answer to encompass those cases as well. Your input is very much appreciated, thank you! Now I know I'm on the right track.

## 1. What is mathematical physics?

Mathematical physics is a branch of physics that uses mathematical tools and methods to study physical phenomena and solve problems. It involves the application of mathematical concepts and theories to understand and explain the behavior of physical systems.

## 2. What is the purpose of writing proofs in mathematical physics?

The purpose of writing proofs in mathematical physics is to provide a logical and rigorous justification for the mathematical models and theories used to describe physical phenomena. Proofs help to establish the validity of these models and theories, and ensure that conclusions drawn from them are sound and accurate.

## 3. How can one improve their skills in writing proofs for mathematical physics?

Improving skills in writing proofs for mathematical physics requires a solid understanding of mathematical concepts and principles, as well as practice and patience. It is important to carefully read and understand theorems and definitions, and to practice writing proofs step by step, using clear and concise language.

## 4. What are some common mistakes to avoid when writing proofs in mathematical physics?

One common mistake to avoid when writing proofs in mathematical physics is assuming the conclusion without providing a valid argument. It is also important to avoid making assumptions without clearly stating them, and to use correct notation and terminology. Additionally, it is important to carefully check for errors and ensure that all steps in the proof are logically sound.

## 5. How can one use proofs in mathematical physics in their research?

Proofs in mathematical physics can be used in research to support and validate theoretical models and predictions, as well as to derive new equations and solutions. They can also help in identifying gaps or inconsistencies in existing theories, and guide the development of new theories and models.

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