4 ms·

Lol... in a previous version of the tutorial, I was using informal language to define vectors as "arrays of numbers," which attracted a lot of negative comments

by ivan_ah 12y ago

Lol... in a previous version of the tutorial, I was using informal language to define vectors as "arrays of numbers," which attracted a lot of negative comments on HN because it wasn't mathematically precise enough. Note the LA book is "Tome II" in the series, and assumes you've read the "Tome I" (no BS guide to math & phys), which means the reader would be equipped to understand more formal math descriptions. But yeah... try giving an explanation for what a vector is that is both: (1) understandable and (2) mathematically precise.

Retra 12y ago
"A vector is a composite quantity that has a magnitude and a direction. If something has a direction component and/or a magnitude component, it can be represented by a vector, and if something is a vector, we can ask for its direction or its magnitude (or both.)"
- ivan_ah 12y ago
  Cool. That works well, if a bit geometrical. I'm going to think about rewording the intro in this way. However, I'd still have to introduce the notion of "coordinate vector" somehow, which begs for the word "tuple" or array, neither of which the right choice.
  - Retra 12y ago
    What I've described is the concept of a vector. You will have to determine the notation, and in setting the notation, and ordered list of basis vector coefficients is most commonly called a tuple, which is a generalization of ordered pair / triple / quadruple, etc.. (Though 'tuple' doesn't usually imply ordered.) The reason 'array' doesn't really work is that arrays are generally understood to be homogeneous in type, while tuples are usually Cartesian products. So if you are using a Cartesian coordinate system, tuple is the ideal term. http://en.wikipedia.org/wiki/Tuple#Type_theory http://en.wikipedia.org/wiki/Tuple#Type_theory
Brakenshire 12y ago
Sorry if it came off a bit harsh. I think the emphasis I would put on all this is variety. All of these books are like different shapes and sizes of tools that you're using to pry open an understanding of the subject. At certain moments you'll want one sort of tool, and at different moments you'll want another. Maybe, on a particular topic, you want to read a chatty, informal explanation, and then you'll want something heavily summarized to nail things in place. Then you might want something really rigorous if you need to go into a lot of detail. Or it might be useful to have a text that scatters explanation of the concepts in amongst the steps to solve practical examples. All of these approaches have value. When I was learning Maths and Physics one of the major conclusions was that there was not enough variety in the material which was available, so when you hit a brick wall, part of the skill was developing an acceptance that you would hit your head against the wall repeatedly until something sank in. Which is inherently frustrating. FWIW I think it's also inevitable that a higher level summary has some kind of slightly leaky abstractions in order to give people a feel for what's going on. I don't see why Maths is any different from Chemistry, or other sciences, in the sense that you need to teach it progressively, at least to some degree. You talk first to a Chemistry student about covalent and ionic bonds, you don't start immediately on Quantum Mechanics and Orbital Theory.
RogerL 12y ago
Try something like "you can think of a vector as an array of numbers. More precisely, [formal definition here]'. If you give somebody an analogy it gives them something to 'hang their thoughts on', if you know what I mean; you read the formalism and try to correlate it back to the analogy. Of course,an analogy is not an equivalence, so there will necessarily be exceptions. You can follow that up with something like "be careful not to assume X from the array of numbers idea; the formal defition states Y, which precludes that". Well, you get the idea.

by ivan_ah 12y ago

Retra 12y ago

"A vector is a composite quantity that has a magnitude and a direction. If something has a direction component and/or a magnitude component, it can be represented by a vector, and if something is a vector, we can ask for its direction or its magnitude (or both.)"

ivan_ah 12y ago
Cool. That works well, if a bit geometrical. I'm going to think about rewording the intro in this way. However, I'd still have to introduce the notion of "coordinate vector" somehow, which begs for the word "tuple" or array, neither of which the right choice.
- Retra 12y ago
  What I've described is the concept of a vector. You will have to determine the notation, and in setting the notation, and ordered list of basis vector coefficients is most commonly called a tuple, which is a generalization of ordered pair / triple / quadruple, etc.. (Though 'tuple' doesn't usually imply ordered.) The reason 'array' doesn't really work is that arrays are generally understood to be homogeneous in type, while tuples are usually Cartesian products. So if you are using a Cartesian coordinate system, tuple is the ideal term. http://en.wikipedia.org/wiki/Tuple#Type_theory http://en.wikipedia.org/wiki/Tuple#Type_theory

Brakenshire 12y ago

Sorry if it came off a bit harsh. I think the emphasis I would put on all this is variety. All of these books are like different shapes and sizes of tools that you're using to pry open an understanding of the subject. At certain moments you'll want one sort of tool, and at different moments you'll want another. Maybe, on a particular topic, you want to read a chatty, informal explanation, and then you'll want something heavily summarized to nail things in place. Then you might want something really rigorous if you need to go into a lot of detail. Or it might be useful to have a text that scatters explanation of the concepts in amongst the steps to solve practical examples. All of these approaches have value. When I was learning Maths and Physics one of the major conclusions was that there was not enough variety in the material which was available, so when you hit a brick wall, part of the skill was developing an acceptance that you would hit your head against the wall repeatedly until something sank in. Which is inherently frustrating. FWIW I think it's also inevitable that a higher level summary has some kind of slightly leaky abstractions in order to give people a feel for what's going on. I don't see why Maths is any different from Chemistry, or other sciences, in the sense that you need to teach it progressively, at least to some degree. You talk first to a Chemistry student about covalent and ionic bonds, you don't start immediately on Quantum Mechanics and Orbital Theory.

RogerL 12y ago

Try something like "you can think of a vector as an array of numbers. More precisely, [formal definition here]'. If you give somebody an analogy it gives them something to 'hang their thoughts on', if you know what I mean; you read the formalism and try to correlate it back to the analogy. Of course,an analogy is not an equivalence, so there will necessarily be exceptions. You can follow that up with something like "be careful not to assume X from the array of numbers idea; the formal defition states Y, which precludes that". Well, you get the idea.