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So you want to study mathematics

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Re: So you want to study mathematics

#341

Post by boppo1 »

technosamay20 wrote: post #210
ghufran_syed wrote: post #16

I went from only having done high school math 10 years ago to completing an MS in math and statistics at my local state university while working in an unrelated field. I would recommend NOT starting with calculus if you haven’t done it, instead, just learn how to do proofs - I used Chartrand “Mathematical proofs” - You don’t need to know any math beyond algebra in order to do that most of this book. If you need to re…

On the other hand I sometimes wonder why Calculus is made a big deal out of. It was probably the easiest among all the parts of maths. You just have to imagine an small unit and how to extrapolate for integration and imagine how to break it down to small parts for differentiation. When I originally learnt Calculus in high school it was couple of days of learning the concepts and the a week of deriving everything from…

Did you multivariable calc in that week? Very impressive.

Re: So you want to study mathematics

#342

Post by ChrisLomont »

wizzwizz4 wrote: post #339
ChrisLomont wrote: post #338

Earlier quoted context omitted.

>That's more to do with the space of the coefficients, I think. I don't think that phrase is a thing in linear algebra. I've gotten a PhD in mathematics - I know both these fields quite well. I stand by my assessment of linear algebra - I've been using pieces form it nearly daily for decades. >Anyone familiar with those basic principles can pick up a new distribution quite easily – and can probably derive new distrib…

> I don't think that phrase is a thing in linear algebra. You're right; sorry for the typo. I meant the span of the coefficients (which is a space). I'm not saying the concept of “rank” is useless, or anything. I'm saying a lot of linear algebra is based on high-level techniques and algorithms, but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra, if you learn /…

>but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra

I'd argue that without a deep understanding of rank and nullity you do not understand linear algebra. Sure, you can lean to move symbols and compute simple things, but that is not much of an understanding.

I'd guess you have not moved into deeper things - then you'll realize you're missing fundamental understanding of linear algebra needed to move on.

It's like claiming one has a fundamental understanding of calculus by being able to solve high school level integrals, but really doesn't understand deep relations between the main ideas of calculus. Sure you can write things down, but there is a major difference between that level of knowledge and what I'd call a fundamental understanding of calculus.

>span of the coefficients

????

Care to link to the thing you're misnaming? I have no idea what you're talking about, and I even googled the phrase.

This is what I mean by using a proper book to learn from.

>Aren't we discussing one?

Nope - the book in question is not offered as a beginner course stats anywhere I am aware of. Care to show one? Calling it one then claiming it is evidence of one is simply circular logic.

>Is that what the book does?

So you have not looked at the book yet are arguing what kind of book it is? That about sums this up.

Re: So you want to study mathematics

#343

Post by boppo1 »

BeetleB wrote: post #233
andrepd wrote: post #157

Earlier quoted context omitted.

Why is it that every time any subject about mathematics comes up there is always a complaint about notation? Your link doesn't even exactly talk about notation, but about pedagogy. Can you be more specific about which notation your consider "arcane"?

> Why is it that every time any subject about mathematics comes up there is always a complaint about notation? The assumption that there is a much better notation is one I tend to see only with the HN crowd. Outside of this group, even people who dislike the notation and/or struggle with math do not claim that a better/simpler one obviously exists.

I really only think calculus notation is an issue. Calc 1 books absolutely demand that you view d/dx as an operator and not a ratio, likewise with the dummy variable and integration symbol. Then later the book teaches about unit normals and it’s implicitly acceptable and treat dx/dz and dy/dz as ratios.

i.e. |u X v| = (1 + (dx/dz)^2 + (dy/dz)^2)^(1/2) = (dz^2 + dx^2 + dy^2)^(1/2).

This was extremely frustrating to me for a while until I accepted that this was how Leibniz did it, so if it’s good enough for him it’s good enough for me.

Re: So you want to study mathematics

#344

Post by wizzwizz4 »

ChrisLomont wrote: post #342
wizzwizz4 wrote: post #339

Earlier quoted context omitted.

> I don't think that phrase is a thing in linear algebra. You're right; sorry for the typo. I meant the span of the coefficients (which is a space). I'm not saying the concept of “rank” is useless, or anything. I'm saying a lot of linear algebra is based on high-level techniques and algorithms, but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra, if you learn /…

>but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra I'd argue that without a deep understanding of rank and nullity you do not understand linear algebra. Sure, you can lean to move symbols and compute simple things, but that is not much of an understanding. I'd guess you have not moved into deeper things - then you'll realize you're missing fundamental underst…

> I have no idea what you're talking about, and I even googled the phrase.

A linear transformation of some n-tuple (let's call it a) to some m-tuple (let's call it b) can be thought of as the sum of an element-wise multiplication of each item of a by an m-tuple coefficient (let's call the coefficient c_i). So b = ∑ a_i × c_i.

The coefficients may or may not be linearly independent. The span (https://en.wikipedia.org/wiki/Linear_span) of these coefficients is a useful property of the linear transformation; I consider it a more fundamental, more useful concept than your "rank".

The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm, and there are lots of theorems that involve it (probably because it was discovered early on in the development of this field), but it seems a rather complicated and unintuitive concept, to me. If you're using “rank”, you need to use a lot of theorems and lemmas and conversions between different representations of things that just don't seem necessary. I am happy to be corrected.

> Nope - the book in question is not offered as a beginner course stats anywhere I am aware of.

That doesn't mean it's not a textbook for beginners. It looks like one, to me.

> So you have not looked at the book yet are arguing what kind of book it is?

It's over 500 pages long. I've scrolled through it a bit, and nothing jumped out as obviously wrong; I haven't read it. (Though I did see a few different distributions named, hence my confusion.)

Re: So you want to study mathematics

#345

Post by travisjungroth »

Someone wrote: post #90
travisjungroth wrote: post #30

Earlier quoted context omitted.

It’s amazing how different the subjects of mathematics are. It’s like the difference between a drum and flute. You listed some of my favorite stuff. Weirdly, when I was 11, my math tutor told me I’d probably really like finite mathematics. She turned out to be right.

> It’s amazing how different the subjects of mathematics are. It’s like the difference between a drum and flute. I think it’s amazing how connected the fields are. It’s almost like “pick any two of analysis, algebra, geometry, number theory, topology, turn one into a adjective and you’ve got a new subject area”. Topological algebra? Check ( https://en.wikipedia.org/wiki/Topological_algebra ) Algebraic topology? Check…

Isomorphs abound!

I think they’re both true. To the musician, the drum and the flute have a sameness. “You make notes in time with them”, the unconscious mastery. To the beginner (me) it’s worthwhile to see their separateness. I shouldn’t think “I don’t like playing instruments.” because of my experience with recorders in the third grade. Don’t let the trauma of high school trig keep you out of graph theory. They certainly feel different.

This is not hollow advice. Yesterday I bought a flute!

Re: So you want to study mathematics

#346

Post by dang »

pphysch wrote: post #96
dang wrote: post #72

Earlier quoted context omitted.

Please don't post in the cross-examining style. We want curious conversation here. This is in the site guidelines: https://news.ycombinator.com/newsguidelines.html .

I don't follow. The author dismissively ctrl-V'd a paragraph with no further explanation, and my response asking for elaboration gets shadow-buried by a mod. What?

Your post wasn't simply asking for elaboration—it was written in the cross-examining style that we particularly want to avoid here because it kills curious conversation.

If you didn't intend to come across like an interrogator trying to back an opponent into the corner, then your comment needed to be written quite differently.

Re: So you want to study mathematics

#347

Post by defective »

Jensson wrote: post #317
radlad wrote: post #314

Earlier quoted context omitted.

> Most everyday people just trust the p values and move on to wondering about confounders they forgot. Perhaps because they never took statistics.

Basically every person who writes papers using statistics has taken a course in statistics. Medical doctors do it, psychologists do it etc. And the statistics course you'd replace calculus with would be even lower level than that, basically worthless to everyone. At least calculus trains you to think about rate of change and slopes, statistics gives you nothing practical. It isn't like people remember those formulas…

People don't need to remember and apply formulas. They just need an intuition that they could be applied and how that might change the appearance of the statistics, as a kind of gauge of reliability. A tool for rationalization.

Re: So you want to study mathematics

#348

Post by eternityforest »

radlad wrote: post #314
eternityforest wrote: post #279

Earlier quoted context omitted.

What would someone other than a scientist use regressions for? I've used one once. I wanted to make a volume control that I thought sounded subjectively even across the range and gave the right amount of control. So I used a polynomial regression calc on a random website and gave it some data points I wanted it to go through. But that's not actually doing a regression, just knowing it exists and computers can do them…

> Most everyday people just trust the p values and move on to wondering about confounders they forgot. Perhaps because they never took statistics.

Or because they aren't scientists and quite likely the original data is hidden for privacy reasons, and also they don't have that kind of free time.

Re: So you want to study mathematics

#349

Post by ChrisLomont »

wizzwizz4 wrote: post #344
ChrisLomont wrote: post #342

Earlier quoted context omitted.

>but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra I'd argue that without a deep understanding of rank and nullity you do not understand linear algebra. Sure, you can lean to move symbols and compute simple things, but that is not much of an understanding. I'd guess you have not moved into deeper things - then you'll realize you're missing fundamental underst…

> I have no idea what you're talking about, and I even googled the phrase. A linear transformation of some n-tuple (let's call it a) to some m-tuple (let's call it b) can be thought of as the sum of an element-wise multiplication of each item of a by an m-tuple coefficient (let's call the coefficient c_i). So b = ∑ a_i × c_i. The coefficients may or may not be linearly independent. The span ( https://en.wikipedia.org…

You concisely illustrate my point. You misuse words so that others that have learned the material cannot understand what you're writing.

Calling an m-tuple a coefficient and then calling the column span the "coefficient span" does not show understanding of the material.

>I consider it a more fundamental, more useful concept than your "rank".

The dimension of the span is the rank. It's exactly why rank is important. The nullity is the dimension of the kernel, and rank + nullity = n.

Saying rank is not important is like saying dimension is not important. The dimension of a vector space is the first and most important invariant that describes the space. Rank is that dimension for the image of a linear transform. It's absolutely fundamental. It's why when describing some linear algebra thing, one usually starts with something like "Let V be a n-dimensional real vector space" or similar. We rarely (bordering on never) write "Let V be a vector space spanned by the following vectors", and we never write "the coefficient span of the transform".

>The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm

No, it is a fundamental property of the linear transformation. Gaussian elimination is but one of many ways to compute it. And no matter what choices you use in your elimination, you will always get the same rank. This is super important - that Gaussian elimination gives knowledge of the linear transformation that is independent of choice of basis or of steps performed. Changing of either vector space changes the numbers in the matrix, and different Gaussian elimination steps may have differing intermediate steps, but the rank is the rank is the rank.

It is also true that the row span = the column span = the rank, which is also not immediately obvious. These are theorems proven in basic linear algebra, and fundamental to claiming to understand basic linear algebra.

That rank shows up in Gaussian elimination is not some artifact or unique thing to Gaussian elimination. Since rank shows up everywhere, when it also shows up in Gaussian elimination shows that Gaussian elimination is doing something fundamental - it is but one of many, many ways that rank pops up over and over in linear algebra.

Rank (and nullity) are absolutely fundamental.

And no one calls that m-tuple a coefficient.

None of this is in the book above. Zero.

A final nice point to illustrate, here [1] is the index to Strang's Introduction to Linear Algebra. Rank occurs more than any just about every other entry in the index.

Here's Serge Lang's book [2]. After the word dimension, rank occurs the most in the index.

Book after book shows that after the concept of dimension, rank is probably the most important concept in linear algebra.

I could go on and on. You cannot claim to understand linear algebra without know how pervasive and useful rank is.

[1] https://math.mit.edu/~gs/linearalgebra/linearalgebra5_Index....

[2] http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_a...

Re: So you want to study mathematics

#350

Post by rawgabbit »

DocTomoe wrote: post #292
ALittleLight wrote: post #280

Earlier quoted context omitted.

But Shakespeare is incredibly useless. People don't even talk or write like that anymore. Any imagined benefit of Shakespeare is just fanciful wishful thinking.

When I was younger, I thought like you did. Now that I get older, the value of culture and an understanding of where it came from, how it developed, makes my life just so much richer. That does not necessarily mean that I need to have memorised the whole corpus of Shakespeare, but understanding a reference when it comes around is adding more layers of meaning to other works. This does not only extend to literature. I…

Shakespeare may have a profound impact in the English speaking world. But from the perspective from any other language than English, Shakespeare is an esoteric subject with limited impact. I do agree that the Christian Bible is of immense importance to European and American culture. Today we blithely dismiss all things religious, but I would argue 99% of modern European/American culture has been affected in some way by Christianity. Pretending it doesn't exist has been a major disservice to students.
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