Vector spaces from the perspective of a linear algebra student

Published December 19, 2020.

On one assignment, I asked my linear algebra students: What is one question you would like to see in future assignments?

Thi Le answered: If the material you learned this week was a food, what would it be?

I asked the question in a later week where we learned about abstract vector spaces. There were so many wonderful answers. This is Claudia Hess's answer:

I would compare what we are learning now to making a sauce. I love cooking and I watch a lot of cooking shows and I listen to Chef David Cheng's podcast. There are rules to making good sauce. If there is a roux, then it must be made first. You must cook the flour long enough so it does not become pasty. This is not an option. If you're making a marinade though, there is no real specific order that needs to take place. I can pour the olive oil or dice the garlic in any order. This is like Rule 1 of the axioms when determining if something is a vector space. According to the recipe, or set in linear algebra, we must determine if order matters according to the rules. For a sauce with a roux, it matters. For a marinade, it does not matter. So, a marinade can be closed under addition while a bechamel is not. The vector space axioms can be compared to any number of sauce making rules. Some chefs say you should season at every step in the cooking process. That is, if I am making a vodka sauce, I should season the tomatoes and onions while they are sauteeing and I should also season the sauce to taste after the ingredients have been blended. Other cooks may say that you need only season when the ingredients are already all together, after they have been blended. I think this is comparable to rule 6[(r+s)v=rv+sv)]. rv+sv is when you season all the parts separately, and (r+s)v is when you season the parts together, assuming the sauce is closed under addition. If I were making something like a bechamel I believe I could season at any point in the cooking process and it would all be the same, so long as I season it properly. If I am making a meat tomato sauce however, this is not closed under addition of seasonings. You should absolutely season the tomatoes and meats separately AND together- it makes a difference in the breaking down of the food and how that changes the flavor profile depending on when and what you season with. Hopefully this makes some sense to you. I am making a bechamel tonight for au gratin potatoes but I do not have gruyere, so I will only use cheddar. I think it should still taste good.

She also said kernel was a dumb name in the final exam when I asked "Find an alternative name for kernel".

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