??! is there such a thing ? only nonsingular square matrices can be inverted. a general mxn matrix may have a “left inverse” or a “right inverse”, but i don’t think your code is computing that.
My feelings is that getting an intuitive understanding is key, so that it becomes part of myself... but I find proofs don't give me this (they show me that it is true, but not why it is that it is true - if that makes sense). Derivations seem important, because then I can re-derive when I forget (though, I must recall the "tricks" of the derivation, and also know the operations used). General skills can persist, if they get ongoing exercise (e.g. methodicalness, care with definitions, close reading, organization to cope with complex and multi-layered problems).
Finally - and awfully - I now think mathematics is like a language, not so much in the sense of communication, but by being full of special cases, exceptions, "abuse of notation". Becoming fluent in a language takes much practice, and may be impossible without special talent. But once achieved, is never fully forgotten, and quickly regained.
"Now this is a matrix you wouldn't want to meet in a dark alley!" - Gil
I can't tell if the author has done a real analysis course before, but if they haven't that's the one they should choose next. If they already have that under their belt they should go to probability or complex variables. I don't see the utility of re-doing calculus or linear algebra if the author is already strong in both.
I appreciate the enthusiasm for math that's evident here, that's great! But if you're learning math the "right" way - by actively engaging with the material - there's only so much of it you can learn at once.
having (somehow) completed many of these requirements for my 18c degree, i would say that analysis is not necessary if your interest is actually applied math. There's a great line in rudin's preface that says that his approach is ~"pedagogically sound at the expense of being logically incorrect," and recommending analysis for somebody that's not looking to mainline a pure math degree to me feels "pedagogically unsound (but logically correct)".
I took analysis and i appreciated it, but i really loved the applied classes in my degree: 18.310, 18.311, 18.781 (theory of numbers) along with algebra 18.701/702. If you haven't taken a higher-level algebra class, it will let you know if analysis is right for you because you'll brush up against the edges of it without (what i consider to be, at least) its hallmark punishing density.
There are other great electives in math at mit, shop around the 18.4* classes and dial in by interest, most of them only require a prereq of 18.02/18.03/18.06 and you can sort of figure the rest out along the way.
Something to be aware of is that for a while 18.310 didn't have a dedicated instructor, so it really was all over the place. 18.311 was also somewhat hastily structured the semester i took it, but it is actually pretty good material.
You may find that after you've done all these classes that you are actually interested in pure math and at that point i would suggest looking at 18.100b (analysis), 18.700 (linear algebra), 18.100c(real analysis), 18.901(topology) and the rest of the "hard math classes," but i really do think that you'll find that the rationale for those classes doesn't click if you haven't taken a few classes like 310 or 701 first.
just my two cents! good luck, have fun!
: this is the actual quote, it's in the preface rudin's principles of mathematical analysis 3rd ed. which is the 'textbook' for 100b.
If you start reading research papers in applied math, there’s a ton of measure theory and functional analysis there.
More generally, both introductory real analysis and introductory complex analysis are assumed basic foundational background for pretty much any kind of research mathematics, applied or otherwise.
I’m also not sure I would recommend trying to self-study them though. Some expert guidance/feedback is pretty helpful for someone starting out.
From my experience (at MIT in the 80s and 90s): The usual student takes 3 technical courses and one humanities course per term. From my own history: I took 18.03 in one term with two EECS courses and a humanities course, and then 18.04 the following term, also with two EECS courses and a humanities course.
I would guess that 3 math courses at one time (and taking nothing else seriously technical) would constitute a pretty full load for most people. Much beyond that probably impacts how much is being truly absorbed for the long term.
My experience (in the 80s) was the same. There were a couple of semesters where I had to take 4 technical courses; I had no desire to repeat the experience. :-)
whatever you intend to learn, consider how you're going to retain it. I'm quite sure that I've forgotten a lot more maths than I know. I'm not sure about the utility of forgotten maths. You may think it rehydrates well, but it doesn't. It won't be as hard to relearn but it'll be hard. So consider, for whatever you learn, are you going to be using it? Are you going to be building upon it? Is it just a tourist expedition? These are all fine reasons but they each have different implications.
I would strongly recommend Chartrand and Zhang's "Intro to mathematical proofs", once you know how to prove things, every other more advanced math class you take will be much easier than it would have been otherwise.
In particular, do the book, including the chapters on "proofs in calculus" and "proofs with real and complex numbers" before doing real analysis - you'll enjoy it much more that way!
I went through most of this book a few years ago, now about to finish my MS in math :)
The reason real analysis is useful is because it's (loosely) a deeper calculus course with proofs. Since probability theory becomes more proof-based (and ventures into measures), real analysis is good preparation for it.
Not necessarily -- a lot of books just take the existence of a unique set with certain properties as an axiom and call it R.
The main topics of basic real analysis IMO are differentiation, integration, (uniform) continuity, compactness, convergence, etc.; how to construct the reals from the rationals is a side point at most.
This could be personal bias as I just don't personally think that the exercise of constructing the real numbers is very interesting.
The obvious benefits are, much of math literature is currently written for consumption by math grad students & profs, so all of this will get a much needed rewrite. I foresee a surge in material like Jeremy Kun’s popular “math for programmers” book that so many of you loved on HN. Much of math lit is imo unnecessarily dense & terse. Speaking as a phd student, I really appreciate atleast 1-2 numerical examples with answers, not just plain theory. Maybe if there was a Linear Algebra for programmers text with some code in python to accompany the theorems, it would be a huge hit among applied ML researchers & programmers. Graduate linear algebra is actually quite hard & very few phd students pass the linalg quals in their first attempt ( Grad linear algebra, not basic shit like matrix algebra, linear maps & decompositions). This material should imo be rewritten ( “dumbed down” ) so regular folk can also sample some of the magic.
In my previous work at an adtech corp, I built a whole bidding algorithm straight from Strang’s linear algebra book (the chapter on Lagrange optimizers). It’s still running, 5 years & counting. I once joked if they gave a nickel out of every dollar that algorithm made to Prof Strang, he’d be a multimillionaire.
I'm going to be starting out on one of these and coding-the-matrix is very python centric (which is super helpful and practical) versus MATLAB in Strang's book.
whether the code is in python or matlab matters little. the code is just a supplement to the main material which is still math. trying to understand that math by reading code is actively counterproductive and actually quite foolish. my professor gave a symposium talk on this topic, so i will use one of his examples -
At 0 minutes you have 0 dollars. suppose each minute you toss a coin. if it’s heads you make a dollar. if it’s tails you pay a dollar out of your pocket. What are the chances you’ll have $30 in an hour ?
This is the sort of classic problem that can be coded up in python or matlab under 5 minutes. it’s just a for loop & an if statement. It won’t tell you much about the actual probability because that number is so small your code will have to run for a long time to get a meaningful answer. Whereas it’s trivial if you just do the math by hand.
not what you thought though.
In the end the only way I could find enough motive to get through a whole applied maths syllabus was to book myself for exams so that I could panic about failing and study.
Then take a course in graduate probability which is based on measure theory. Good authors are Loeve, Neveu, Breiman, Chung, among others. So, learn about the cases of convergence including almost sure convergence, a good version of the central limit theorem, proved carefully, the weak and strong laws of large numbers, ergodic theory, and martingale theory.
Learn stochastic differential equations.
Learn some potential theory.
Then do some work in optimization and optimization under uncertainty.
I didn't like either Royden or Rudin. I really like Folland's book "Real analysis: modern techniques and their applications." His Fourier analysis book is also pretty good IMO.
I just never could see much utility in complex variables: Complex valued functions of real variables. Sure. Complex valued measures? Sure. Fourier theory making a lot of use of complex numbers? Of course. A vector space where the field is the complex numbers? Certainly. Hermitian and unitary matrices with complex numbers? About have to like those due to the fundamental theorem of algebra, that is, roots of polynomials and, thus, complex eigenvalues. Functions of a complex variable? Never could see the utility.
A good course in statistics, e.g., with sufficient statistics, needs graduate probability, and with a grad probability course can do elementary statistics easily in the footnotes. E.g., the proof I worked out for the Neyman-Pearson lemma is quite general but based on the Hahn decomposition based on the Radon-Nikodym theorem (Rudin gives von Neumann's cute proof) in measure theory. The grown up approach to conditional probability is based just on the Radon-Nikodym theorem of measure theory.
18.1XX Analysis / Calculus
18.6XX Probability and Statistics
18.7XX Algebra and Number Theory
18.8XX Just project lab
18.9XX Topology and Geometry
6.82x tend to be systems classes like networks or OS.
But 17.828 isn’t like an intro to constitutional law and 6.42 isn’t Causes and Prevention of Software Project Failure.
18.XYZ where Z >= 5 are grad classes.
X = 1 is Analysis
X = 2 is Mathematical Physics
X = 3 is Applied Mathematics
X = 4 is Theoretical Computer Science
X = 5 is Logic
X = 7 is Algebra
X = 9 is Topology
Studying applied math gives you the skills of being able to approach problems in a logical, analytical, and above all PERSISTENT way (because let me tell you will you ever have some late nights banging your head on the desk trying to wrap your brain around these problems). These skills are crucial in any field where you need to create mathematical representations of a system - or break apart and understand someone else’a models.
Also - partial differential equations are just cool! They are like a mathematical rubick’s cube. Solving them will legitimately make you smarter.
In particular, I'm hoping to self study the Structure and Interpretation of Classical Mechanics, but can't find the answers anywhere.
I don't know anything about music theory, but if a course can be completed in three hours, it's a soft course.
I don't think math works like this. I specifically decreased my pace when doing math courses that I hadn't done in undergraduate and I am glad that I took that approach. I now see any problem (given enough time) as more of a research question than just a box to tick.
> He taught me “the standard pace is for chumps” — that the system is designed so anyone can keep up. If you’re more driven than most people, you can do way more than anyone expects. And this applies to all of life — not just school
I ended up doing honours in bioinformatics, but then took a year to do third year pure math (and the honours course in lattice theory and topology). After this I ended up doing what I really wanted to, which was an MSc in pure mathematics.
I also did probably the first 1/3 of Strang's course (even though I had already done linear algebra formally in my original course). I think online math courses are a mixed bag and it could even be that it works better when you have an idea already of why you want to do it and the applications thereof. Applications need not imply applied math—you can apply mathematics to other branches of mathematics.
The more difficult thing for me is to independently find research topics and subtopics. Mentorship in mathematics is very important.
EDIT: Brain health also derives from a healthy social life, I forgot to add that to the list involved. Having friends that are interested in learning the same things as you, and discussing it with them, help you to develop self-checking techniques.
Also, Alan Kay pointed out in a speech in the last few years that our economy is doing really poorly right now producing researchers, and people who understand research, and you really can't get there without a good education.
I'll go one further and suggest that 'Transfer', which is the application of understanding of one topic to another, is one of the prime sources of what we consider 'innovation' or 'paradigm shifts'.
I feel as if that should be enough proof that education is an end to itself, but I need to be really clear: continuing education doesn't need to have a goal, and I would strongly discourage you from adding one. I go to the gym all the time without goals. I go for walks everyday, without goals. Each of these items are basic maintenance.
Learning new things _well_ is basic maintenance for your brain. If you want to do continuous learning then you should install Anki, get a notebook and learn Cornell notes, then go out and take a quick class, lecture, or study material.
If you are learning something new, the fastest way to learn and retain the material is to move to self-testing as quickly as possible. I can go over this in more detail, but perhaps the first class I would recommend would be "Learning How To Learn" (Coursera), which produced a serious attitude shift in me about learning and what I am capable of as a human.
A suggestion: 18.01 and 18.02 are typically taken in the first two semesters of freshman year (and many take 18.03 the first of sophomore year, or even the semester before). As you're well past the high schoolers who push up at MIT, I suggest 18.014 / 18.024 / 18.034 for more theory (those weren't available when I was a student).
It's funny: when I was an undergrad Math was quote popular because it had so few requirements and on your page it still looks that way (compare to course 6). I wonder if they same dynamic still applies?
esp for applied math it's invaluable compared with python or matlab.
Juno is a nice ide, it's gotten a lot better over the last two months.