- I updated my notes on Vector Spaces. I have added more notes to my abstract version of the even-odd functions problem. Admittedly, this problem is peculiar—some parts are straightforward, while others require you to discover and formulate new statements. I assign such tasks because exploration and discovery are integral to learning. If you fail to arrive at discoveries for which you are confident make mathematical sense, document your explorations and reasoning; partial credit will be awarded for your explorations and reasoning. The same notes with cropped pages.
- I updated my notes on Vector Spaces. I corrected the errors that you pointed out in Discussions on Canvas. The same notes with cropped pages.
- I updated my notes on Vector Spaces. I corrected the errors that you pointed out today and added some more content that I presented in class. The same notes with cropped pages.
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The reading homework for Thursday:
- Section 1A in the textbook: Review of complex numbers, the concept of a list (I prefer to call lists tuples), the space $\mathbb{F}^n$ (Notice that in this book the author uses the old-fashioned bold face capitals to denote the scalar fields. I prefer to use, so called blackboldbold font, which seems to be more common nowdays.)
- Section 1B in the textbook: Definition of a vector space and exercises.
- If you find an exercise that you cannot solve, please report them in Discussions on Canvas.
- (updated) My notes on Vector Spaces. The same notes with cropped pages.
- My notes on Complex Numbers. There are 11 exercises at the end of my notes. Almost entirely the content of my notes is mentioned in the Common Core State Standards for Mathematics. Some of the exercises are very similar to the exercises mentioned in the Common Core State Standards for Mathematics.
- My notes on Mathematical Logic. The same notes with cropped pages.
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I recommend that you look into learning
LaTeX which is a superior typesetting system for writing math papers. Not only that LaTeX is the best way to write mathematical content on computer, but it is the best way to display math on the Internet. All the mathematics that you see on my website is written in LaTeX. All the mathematics that you see on Wikipedia is written in LaTeX. In fact, if you go to a mathematics page, on the line below the title, next to the Read button, you will find the Edit button. Clicking this edit button will take you to the code for that webpage. You will see a lot of LaTeX code there. In fact, if you need exactly that formula, you can copy and paste it in your document.
- I created Getting Started with LaTeX page to help you with this.
- Here is a simple LaTeX sample document in which I prove an interesting inequality.
- If you need help starting with LaTeX, feel free to ask me for help. It is as important as learning one nice piece of mathematics. And I want to help you with that, not only because that is my job, but because I deeply believe that both math - through learning rigorous thinking - and professional writing skills - supported by rigorous thinking - will help you in your life more than anything else.
- I have noticed that many students use the online LaTeX editor Overleaf. I do not have any experience with Overleaf, except that I have seen that many students use it with nice results.
- LaTeX is just like a computer program that compiles into your mathematical document. As you probably heard ChatGPT is exceptionally good at writing code in all kind of different languages. So, it is good at writing LaTeX code. If you are confused on how to do a particular task in LaTeX, you can ask ChatGPT for advise. All advise that I obtained was superb. Not only in LaTeX, but in Wolfram Mathematica as well. Sometimes ChatGPT is sloppy, and makes trivial mistakes, parenthesis not matching and stuff like that. It helps to write very detailed user prompts.
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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.
- It is tempting to make LaTeX a mandatory tool for creating submissions in this class. However, I do not appreciate having mandates imposed on me; I prefer the freedom to think my own thoughts. An important part of a teacher's role is to encourage students to think their own thoughts, and imposing a mandate could contradict that spirit.
- Professor Sheldon Axler, the author of Linear Algebra Done Right has made PDF file of the fourth edition of this popular linear algebra book available free of charge. We are taking advantage of this generous gesture and we are using this book this quarter.
- The information sheet
- Some relevant Wikipedia links:
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Before briefly reflecting on the history of Linear Algebra, I want to celebrate the history of mathematics with one long sentence:
Throughout history, human civilizations continue to develop and share mathematical knowledge. Succeeding civilizations recognize and admire the contributions of the preceding ones. Prior mathematical creations serve as a foundation and inspiration for further advancements, all in unity, giving mathematics the true spirit of a collective endeavor of the entirety of humanity through history, the present, and the future.
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What is the oldest linear algebra problem?
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Clay tablet VAT 8389 from the Old Babylonian period, from 2000 to 1600 BC, contains what is believed to be the earliest word problem that can be interpreted as a system of linear equations:
A translation of this word problem into a system of linear equations is as follows: \begin{alignat*}{4} &x_1 & &\ + &x_2 & = 1800 \\ \tfrac{2}{3} &x_1 & &- \tfrac{1}{2} &x_2 & = \phantom{1}500. \end{alignat*} -
Problem 40 of the Rhind papyrus which is dated to 1550 BC is:
Denote by $x_1$ the smallest number and by $x_2$ the common difference. After simplification the above problem translates into the following system of linear equations: \begin{alignat*}{5} 5 &x_1 & & + 10 &x_2 & = 100 \\ \tfrac{11}{7} &x_1 & & - \phantom{1}\tfrac{2}{7} &x_2 & = \phantom{10}0. \end{alignat*} -
Most importantly for us, the oldest known treatment of systems of linear equations from antiquity which resembles the methods that we will use in this class is in Chapter 8 of the Chinese textbook Nine Chapters of the Mathematical Art which is at least 1800 years old.
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Clay tablet VAT 8389 from the Old Babylonian period, from 2000 to 1600 BC, contains what is believed to be the earliest word problem that can be interpreted as a system of linear equations: