Summer of Math Exposition

Hilarious, very memorable presentation style. Clear explanation and definitions. Learned what Co- and contravariance mean for vectors! The final bit on basis was a few steps too far removed from the rest of the article, though, so it concluded on a muddled/confused note.

As a college student, I came across co and contra vectors while studying Electromagnetism (marrying it with relativity). And I think your explanation for the co and contra part was really amazing! I think it was very easy to follow for me. And the way both the vectors change with a basis transformation is now very clear to me! Thank you for that!

That being said, I think having a more clear indication (other than small or caps) for price and shopping list could have been easier to follow. The notation itself could have been explained more swiftly I think, although it is understandable that Matrices are just something that takes getting used to while using it for computations to first understand them.

I liked the energy with which you wrote this article, the playful drawings, and the main idea: teaching how vectors can be interpreted as solutions to a system of equations, as algebraic objects in their own right, and as geometric objects, and using this to define covectors and contravectors. Your organization of the main ideas was straightforward and reasonable!

I will say though that several things make this article feel more like your personal notes than like an exposition, and made it feel more like your personal notes:

• the handwritten prices of food and equations were a little difficult to read (sadly, here good handwriting is important!)
• the texting-style language distracted from the exposition (for instance, the first few sentences have playful remarks like “okok let’s start w fruit :>” and “simple nuff :3” which make me forget about what you’re trying to explain in the first place)
• the typed text was hard to read in a similar way that your handwriting is (eg. “This^ on the left is called a 1x3 matrix and this^ on the right is a 3x1 matrix” doesn’t exactly align with the image above it). I really don’t want to discourage you from using playful language and drawings, especially if this is how you take notes—your style is very charming and makes it clear that you are making these ideas your own, so please keep doing this for yourself!

But if you are intending this for a wider audience, I think the most important thing to do is to see how you can answer a few questions:

• Why do you want to answer the question “huh what’s a covector” in the first place? who would care about an answer in the first place, and why would they care?
• If your intended audience is seeing vectors for the first time:
  • Why would they know that the solution to a system of equations can be interpreted as matrix multiplication? (You kind of answered this question by demonstrating how matrix multiplication works, but I think it’s still not very clear because you say “don’t worry too much about the specifics of matrices right now, just know what it represents. :>”; at that point the reader might not even know what matrices represents and be confused)
  • Why would they know that matrix multiplication can be interpreted as actions on vectors in some n-dimensional space? (This question isn’t explicitly answered in your article)
  • Why would they know that vectors can be interpreted as solutions to a system of equations? (This also isn’t explicitly answered in your article)

I hope these questions are helpful! This is a very difficult topic to introduce to someone for the first time, but well worth it to try to explain deeply.

I really enjoyed the striking layout, the simple informal language, and the conciseness of the exposition. I am a bit more mixed about the visuals: I like that you didn’t go for super polished formulas and diagrams that some entries overuse, your diagrams were interesting to look at, and also I’m guess that having visuals that didn’t take a huge amount of time to make meant that you had more time to focus on the text. But I was wondering if you could have had maybe a couple of visuals that were beautiful to look at.

I think it would have been better if you’d discussed some of the applications of covectors. In the current article it’s kind of a curiosity, like, instead of writing the value as a sum you can write it as a vector-covector product, but can we use that for something? Or is it just a notation shorthand?

As an explanation of computing the vector-covector product I think it’s a really nice intro. I would definitely share it with someone who was new to the area. And I really liked the questions at the end, they were great. They just felt a bit rushed maybe? Perhaps there’s a way to fit them more naturally into the text?

Could be written cleaner and/or more structured. But the actual explanations of co- and contravariant vectors was really good!