The context of all questions in Problem 5 is the vector space
In part (a), one should aim for the simplest possible proof. Applying heavy theorems from Mathematical Analysis to polynomials seems inappropriate. One-line proofs that merely cite a major theorem should, in my opinion, be avoided. If using such a theorem is absolutely unavoidable, one should review and cite it in full detail. Proofs from first principles are always preferable.
We use the following notation:
The vectors in abstract vector spaces are denoted by lowercase Latin letters, such as
In displayed formulas, we will use the following logical symbols:
In a preceding ■ item we introduced the isomorphism
Let
Below, I present a remarkable example of a proof using the Principle of Mathematical Induction. This proof appears as Example 4.5 in the book:
A student recently stopped by my office to discuss Problem 3 on Assignment 1. In the narrative accompanying this problem, I hint that polar coordinates might be a useful tool.
Polar coordinates often prove invaluable. For example, working with complex numbers, via Euler's identity, see the end of the post today.
Over the years, I have noticed that students often approach polar coordinates with a certain degree of overconfidence. However, this topic warrants careful attention. As is often said in discussions about Large Language Models: Attention is All You Need.
For the proof of the Polar Coordinates Theorem, let
I. First solve the equations
II. With the result from I, the point
III.
If
IV.
If
V.
Since
This completes the proof of the Polar Coordinates Theorem.
Since I spent all this space on the details of polar coordinates, let me recall that in the context of complex numbers there is standard terminology for all four real numbers
|
Terminology | Notation |
---|---|---|
the real part of |
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the imaginary part of |
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the modulus of |
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the principal argument of |
The website Argument (complex analysis) - Wikipedia distinguishes the principal value of the argument, denoted by
I am proud to promote the simple formula for the principal value of the argument:
For
Since I often rewrite, to celebrate thinking, writing, and teaching, I created my own saying
I came, I thought, I wrote, I taught, I thought more, I rewrote.
You:
Can you please write a complete LaTeX file with instructions on using basic mathematical operations, like fractions, sums, integrals, basic functions, like cosine, sine, and exponential function, and how to structure a document and similar features? Please explain the difference between the inline and displayed mathematical formulas. Please include examples of different ways of formatting displayed mathematical formulas. Please include what you think would be useful to a mathematics student. Also, can you please include your favorite somewhat complicated mathematical formula as an example of the power of LaTeX? I emphasize I want a complete file that I can copy into the LaTeX compiler and compile into a pdf file. Please ensure that your document contains the code for the formulas you are writing, which displays both as code separately from compiled formulas. Also, please double-check that your code compiles correctly. Remember that I am a beginner and cannot fix the errors. Please act as a concerned teacher would do.
This is the LaTeX document that ChatGPT produced base on the above prompt. Here is the compiled PDF document.
You can ask ChatGPT for specific LaTeX advise. To get a good response, think carefully about your prompt. Also, you can offer to ChatGPT a sample of short mathematical writing from the web or a book as a PNG file and it convert that writing to LaTeX. You can even try with neat handwriting. The results will of course depend on the clarity of the file, ChatGPT makes mistakes, but I found it incredibly useful.