How to cover undergrad?

I am going to cover all of undergraduate maths and science on this website, to a high standard.

I would like your help, and several people have spoken to me directly about this.

What are people’s thoughts in general, what are people’s thoughts on the current Linear Algebra section?

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Ideas for possible sections based off oxbridge/warwick first&second year pure content, Imperial probably similar:

  • Real Analysis (in \mathbb R, (first year and bits of second year) \mathbb R^{m \times n} and \mathbb R^n (second year))
  • Complex Analysis
  • Elementary Number Theory
  • Algebra (Group Theory, Ring Theory, Field Theory)
  • Vector Calculus
  • Geometry (classical and modern [including spherical, hyperbolic, etc.])
  • Differential Equations/PDEs
  • General Topology
  • Variational Principles/Calculus of Variations (bit of a niche one)

I think that basically covers everything important in first/second year.

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This is excellent. I wonder what the best way to do it would be. Ideally there would be a consistent framework across topics, but huge scope for users to contribute in their own way so there is a really rich set of content that can improve indefinitely.

I was considering breaking a topic into elements (like I started with Linear Algebra) with definitions + theorems. These should be consistent across universities. I propose ‘CleanSlate’ posts these.

But then aside from this, the flow of the course, examples and exercises and new problems could come from the community.

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Sounds good to me

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I have access to quite good resources on these topics but if anyone else also has some, feel free to share

https://dec41.user.srcf.net/notes/
^ lecture notes from Cambridge lectures.

https://www.maths.cam.ac.uk/undergrad/examplesheets

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I need to add a ‘I love this’ button

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I am proceeding with Linear Algebra Definitions and Theorems now. If another module takes someones fancy they should proceed with that and message me if you are blocked by some site functionality. I can make categories for example.

We should expect to have to iterate a lot and learn what works and what does not work. Therefore a critical eye is very helpful and appreciated.

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I like the CleanSlate format for this: when you split notes up into tiny chunks (definitions, theorems, examples, etc) it means that each one can be individually commented on, and so people can ask questions about very specific things in exactly the right place

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These are my thoughts as well. The university course, but help that is an order of magnitude faster than the tutorial system.

You could never run out of exercises, if you want more, because you could just make a request and the community would come through for you.

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Yeah, the first port of call after university example sheets imo should be classic textbooks. (eg. Munkres for Topology, Rudin for Analysis)

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Is that because the flow of these text books works really well in your view?

BTW max post length is 150,000 characters. So there is scope to write chapters of a course on Clean Slate. Perhaps definitions, theorems, etc are broken into their elements, but then referenced in the main body of text.

Maybe have an index? Something like:

Section 1

  • Definition 1
  • Definition 2
  • Theorem 1

etc.

Maybe we could also have narrative and motivation inbetween too.

Yes I think that’s important. You could have a ‘main’ post or a post per chapter that provides the narrative and then links to definitions and theorems. There is a 150k character limit so I think there is scope for this

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Personally, the most useful thing for me is the narrative and motivation. I dislike being provided with theorem, even when proved, out of the blue. I need to see how the author came up with the thought. What motivation was there to conjecture said theorem is true?

This has always bothered me with the definition of the determinant of a matrix. With the best of my efforts I have still not found the answer to “How did one come up with the concept of the determinant?”.

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I strongly agree. We can sort of have our cake and eat it. Have a main post that contains everything so the explanation and flow is there, but also have a collection of all definitions and theorems. Latter is pretty useful for revision in my view.

Linear Algebra will soon contain the determinant definition and you can ask exactly that question in the replies to it. If someone produces a good answer, perhaps that makes it into the core course

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It depends where we place ourselves. Do we aim to replace a textbook, ie. make it so that students can use this site in place of a textbook, or just try to supplement a student’s study with worked problems.

I try to add footnotes to my posts to give interesting insights and hints towards broader implications. (forgot which one it was - but the question which had you classify all orthogonal 2 \times 2 matrices which in turn classifies all isometries in \mathbb R^2 for free, at least once you establish some intermediate theorems)

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Forgot to reply to this - it’s just that they’re a good source and easy to run through.

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