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Linear Algebra - Foundations to Frontiers

edX · The University of Texas at Austin · 6 HN points · 10 HN comments

Learn the mathematics behind linear algebra and link it to matrix software development.
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I'm e-learning Linear Algebra right now to have a good math foundation for Machine Learning.

I was a History and Sociology major in college - so I didn't take any math.

If you are like me, and working off an initial base of high school math, I would recommend the following (all free):

Linear Algebra Foundations to Frontiers (UT Austin) Course: Comments: This was a great starting place for me. Good interactive HW exercises, very clear instruction and time-efficient.

Linear Algebra (MIT OpenCourseware) Course: Comments: This course is apparently the holy grail course for Intro Linear Algebra. One of my colleagues, who did an MS in EE at MIT, said Gilbert Strang was the best teacher he had. I started off with this but had to rewind to the UT class because I didn't have some of the fundamentals (e.g. how to calc a dot product). I'm personally 15% through this, but enjoying it.

Linear Algebra Review PDF (Stanford CS229) Link: Comments: This is the set of Linear Algebra review materials they go over at the beginning of Stanford's machine learning class (CS229). This is my workback to know I'm tracking to the right set of knowledge, and thus far, the courses have done a great job of doing so.

I agree, Strang's Linear Algebra course is excellent. I worked through the entire course in March/April this year.

I just completed my final exam at CMU in their graduate intro to ML class (10-601). Having gone through the LA course was essential to my success. But equally important (if not more) to ML is a solid foundation in probability.

Out of curiosity, do you feel you can compete with people who have advanced degrees in more quantitative sciences?

Although I'm in the process of plugging several holes in my own math education, I don't believe I'd be able to get any interesting, ML related jobs. I also don't see myself able to perform well in comparison, given that I lack the mathematical intuition one builds after several years of (almost) daily practice.

(I hope I don't sound discouraging, relearning math has been quit fun so far and made me able to understand more of everything).

The direct answer is - not at the research level but yes, in terms of application as the field matures and abstracts over time.

There are also adjacent jobs to ML engineer (product management and so on...).

did you find LAFF too mathy? i got turned off by the mathyness of it and i quit in 2 weeks. does it get any better? All the math notations and lines got so dry that i vapourised trying to understand.
I didn't think about it during the time. It's a fair comment, and probably true.

What it did really well (for me) was integrate HW with each lecture video, and start at a really basic foundation. It took me from 0 -> something.

What's the difference between learning and e-learning? Is the latter faster?
Orders of magnitude cheaper.
Good recommendations. In addition to the UT, MIT and Stanford courses you recommend above, for developing your visual intuition, 3Blue1Brown's Essence of Linear Algebra video series is second to none. [0]

Another good one is MathTheBeautiful [1] by MIT alum Pavel Grinfeld [2]. He approaches Linear Algebra from a geometric perspective as well, but with more emphasis on the mechanics of solving equations. He has a ton of videos organized into several courses, ranging from in-depth Intro to Linear Algebra courses to more advanced courses on PDEs and Tensor Calculus.

Esp note his video on Legendre polynomials [3] and Why {1,x,x²} Is a Terrible Basis:

Gilbert Strang was Greenfield's PhD advisor: Pavel has a clear and precise teaching style like Strang, and he makes reference to Prof's Strang and his MIT course from time to time.

NB: Prof Strang has a new book Linear Algebra and Learning from Data that just went to press and will be available in print by mid Jan 2019. A few chapters are available online now, and the video lectures from the new MIT course should on YouTube in a few weeks. [4]

[0] Essence of Linear Video Series (3Blue1Brown):

[1] MathTheBeautiful



[4] MIT Linear Algebra and Learning from Data (2018)

Don't forget to review calculus as well. Khan Academy is a good start for learning about single variable calculus (, but their content on multivariable calculus is a bit lacking (neural networks / deep learning use the concept of the derivatives and the gradient a lot). A good supplement for multivariable calculus would be Terence Parr and Jeremy Howard's article on "All the matrix calculus you need for deep learning":
Thanks - I am doing that as well! I've been using MIT OpenCourseware for single variable calculus (and will do the same for multivariable). I fenced the parent post to Linear Algebra to not go too far away from the OP.

I will certainly check out the Terrence Parr / Jeremy Howard site, and am super familiar with Khan Academy.

I'm coming to the end of my first year (6 year part time) Comp Sci course and have seen that we have options for AI and Machine Learning modules in future years. Where should I go to find something like a list of what I should be brushing up on, or learning completely from scratch, in order to not fall flat on my face during those type of modules.

I understand there are very set starting points in math subjects because concepts build on one another but I don't know what I should be starting with and where to go afterwards.

I'm going to plug Calculus: Single Variable from the University of Pennsylvania on Coursera (

This was the best Calculus course I've taken online.

When you say you’re going through the MIT OCW calculus courses are you watching the videos or also doing practice problems?

What other calculus resources are you using?

Watching videos, reading the (indicated portions of) the text, doing practice problems, eventually exams - relying on the resources provided in the OCW site.

To be transparent - I just started the calculus class. I finished the UT Austin Linear Algebra class two weeks ago, and am 7 lectures + readings + 2 problem sets in on the MIT Linear Algebra class and 3 lectures in on the Calculus class.

I came across-Calculus Made Easy by Silvanus Thompson,on someones twitter feed. Published in 1910 and far less scary and far more interesting to read than a lot of math text books.

Similarly, Stroud's "Engineering Mathematics" takes you right from addition all the way up to Fourier transforms... a fantatsic book.
“Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.”

I really wish technical books were still written like this. Though if Thompson posted this on HN as a comment he probably would have been downvoted.

There is also a web goodlooking version, discussed previously in HN:
This should be your first calculus book if you are learning from scratch. Much better than those 1000+ page behemoths colleges use.
That's a pretty good list, here are some things I'd add.

Amazing js visualizations/manipulatives for many LA concepts:

LA Concept map: (so you'll know what there is to learn)

Condensed 4-page tutorial: (in case you're short on time)

And here is an excerpt from my book: (won't post a link to it here, but check on amazon if interested)

Wow - nice stuff. I'm a fan of the reference map.
3blue1brown has a great youtube series on both Calculus and Linear Algebra that provide excellent intuitive backing for the concepts in both areas.
Would be interested to hear why you’re studying machine learning. Do you see important problems you think it can solve, are you looking to make more $$ as an ML data scientist, or just generally interested in stats/data?
Part of a broader effort - I committed to learning to code about 3 years ago. At that point in time, I didn't really know why I was doing it... really out of curiosity. I kept going because it was addicting, and really an antithesis to my day job at the time (investment banking) - which I felt was corporate / bureaucratic and unintellectual.

That said, eventually, I want to start a startup. I'm building out small side projects now. I'm generally comfy with web + mobile dev, and I wanted to upskill in a "newer" technology that was more "mathy".

Thanks for sharing, good luck on your journey
I bet you could flow-chart your entire firm into a process that's mostly automated. Something like wealthfront, maybe?
I just love the MIT linear algebra course with Gilbert Strang. Awesome teacher
A bit weird to add a negative review, but here goes:

Is _not_ a good introduction. The instructors are all over the damn place, and you will spend much of your time finding better explanations from other sources. Wish I hadn't started with this. On the plus side, you will get a certificate at the end.

Indeed weird, because the course you mentioned is actually excellent. However, it was designed for people who had (somehow) already seen the subjects in an abstract and unapplied setting (such as a math class at uni). They refresh or refocus the subjects with a geometric intuition and with some concrete applications in mind; which I found quite useful and beautiful. This class is more like a more developed version of 3b1b videos on LA.
It's funny, because as I was reading your comment, I was thinking of 3b1b. He's doing great work by visualizing abstract concepts, but I think what he does mainly helps people who have already gone through the material. If it's your first time encountering the topic, you'll likely feel lost or not see the point.

What 3b1b does still brings a lot of value, so I don't want to take away anything from his work.

If you care about anything that runs on a computer, linear algebra is one of the best maths.
> This course [Strang] is apparently the holy grail course for Intro Linear Algebra.

I haven't watched his lectures, but I TA'd a linear algebra course that used his text book, and strongly disliked his presentation. I've heard that's a fairly common reaction actually - it's one of those love it or hate it books. I'm bringing it up because if you (or someone else reading this) turn out to be in the group that doesn't love it, you should not give up on loving linear algebra! You are definitely still allowed to have a different 'holy grail course'!

Gilbert Strang is probably the best teacher on videos, up there with Dan Boneh.
Awesome input! Learning isn't linear (tee-hee...)
Where’s the love for Lax?
Page after page of mathematical insights and delights! I've never had the opportunity to work through it systematically, but have frequently read excerpts and have never been let down. I would expect nothing less from a figure so great as Lax!

It's worth pointing out in the context of this discussion that the book is, by the author's own design, not an introduction to linear algebra. It is a second course that Lax used to teach his advanced undergraduates and beginning graduate students at the Courant Institute. For example, OP with a high school math background will surely be very puzzled by page two, when a linear space is defined as a field 'acting on' a group. Which is, i think, the 'right' way of thinking about the algebraic structure, in the sense that it greatly simplifies all the intricate moving parts of linear algebra. Anyhow, I second your recommendation!

Author here. I made this demo and a related matrix-matrix multiplication demo [1] back in 2015 for Robert van de Geijn's Linear Algebra: Foundations to Frontiers MOOC class [2]. In the light of Spectre attack and recent browsers' changes to reduce precision of timers, I remembered of this project, and decided to check if it still works now, 3 years later. Surprisingly, it still works well!

The source code is available on GitHub [3].




Could you make the website viewable without JavaScript?

Edit: I think the downvotes are unjustified. For clarification, if that wasn't clear by context, I don't expect to get the JavaScript test results from my computer while viewing the website without JavaScript. Demanding that would obviously be nonsense. Rather than that I assume there is information on that website that is interesting to read even without using JavaScript personally. Or is using JavaScript now a requirement to learn about JavaScript?

I guess you'd have more luck if you had asked "can you make some examples accessible for those of us who don't run JS?".
I didn't know what to expect to see. I vaguely assumed to see some text and data. That's why I carelessly formulated my question like I did. Still, I think the difference is small and my question wasn't extraordinary.
Is there a particular reason why you can't enable JS yourself? Metered connection? Low-end machine?
I consider it an unacceptable form of code deployment. It's unsafe in the computing sense, and leads to a ecosystem where users are less and less in control of the software they use.
Metered wouldn't really matter would it? Scripts are still downloaded, right? (never had JS off, because the web)
Metered could matter - it's very easy to disable downloading of any non-inlined JS (like with uBlock)
I use uMatrix, this site just shows up as a white page until I allow it to load some JS from a third party. Once I allow the JS, I see a graph that gets built without any explanation of what that graph means.
There's only a graph of the results, so you are not missing any other content.
Thanks for the info.
Because they are written by mathematicians. In my case, when I have learned a mathematical topic, the intuition becomes obvious and the derivations/proofs seem to be much more important for gaining a complete understanding. I have gone up against texts with complete bewilderment, only to come back after gaining the intuition and found the extensions of the core premises and proofs provided by the text to be highly enlightening.

Great math teachers understand the need to teach intuition. He wasn't a math teacher, but I think Richard Feynman is the pinnacle of this. See [1] to see how he expresses intuition about physics, and his Red Books[2] for how he teaches mathematical physics with all the qualities I believe makes a great maths text for students.

Also, there's a linear algebra MOOC which also teaches great intuition before delving into proofs and heavy detail [3]. I mention these examples because they are exemplars of this idea of teaching intuition.




> Richard Feynman is the pinnacle of this

There really needs to be a version of the Feynman lectures for mathematics.

Although, this is what Arnol'd has to say [1]:

"Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap... In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences."


That rant is a bit ridiculous to post here - the foundations of computer science is largely the result of mathematicians who weren't particularly interested in physics.
Well for one Turing was certainly somewhat interested in physics. From his WikiP page:

In 1928, aged 16, Turing encountered Albert Einstein's work; not only did he grasp it, but it is possible that he managed to deduce Einstein's questioning of Newton's laws of motion from a text in which this was never made explicit.

Then there was von Neumann and several others. If not interested then at least well educated in physics.

The foundations of computer science are trivial from the mathematics standpoint. I would not call them "results of mathematicians".
Computer science is far from trivial. In fact, it's arguably the only part of mathematics that is provably non-trivial.
> There really needs to be a version of the Feynman lectures for mathematics.

I asked the question on math.SE:

In search for books like these in the past, I found "Understanding Analysis" (Stephen Abbott). I went through the first two chapters and I liked it. It is written in a narrative which is both entertaining and instructive. He explains the problem, why is it relevant, ways of approaching it, etc. "It is designed to capture the intellectual imagination."

From the preface: "This book is an introductory text. The only prerequisite is a robust understand- ing of the results from single-variable calculus. The theorems of linear algebra are not needed, but the exposure to abstract arguments and proof writing that usually comes with this course would be a valuable asset. Complex numbers are never used.

The proofs in Understanding Analysis are written with the beginning student firmly in mind. Brevity and other stylistic concerns are postponed in favor of including a significant level of detail. Most proofs come with a generous amount of discussion about the context of the argument. What should the proof entail? Which definitions are relevant? What is the overall strategy? Whenever there is a choice, efficiency is traded for an opportunity to reinforce some previously learned technique. Especially familiar or predictable arguments are often deferred to the exercises.

The search for recurring ideas exists at the proof-writing level and also on the larger expository level. I have tried to give the course a narrative tone by picking up on the unifying themes of approximation and the transition from the finite to the infinite. Often when we ask a question in analysis the answer is “sometimes.” Can the order of a double summation be exchanged? Is term-by- term differentiation of an infinite series allowed? By focusing on this recurring pattern, each successive topic builds on the intuition of the previous one. The questions seem more natural, and a coherent story emerges from what might otherwise appear as a long list of theorems and proofs."

Feynman learned mathematics from a series of self teaching books published in the 1940's suffixed "...for the Practical Man" and prefixed with Arithmetic, Algebra and Calculus. I have the full set and this is a rather good solution to the problem. They teach you insight and how to think about things as well as the mechanical aspects. This is IMHO a well solved problem if you don't mind skipping more modern abstractions such as limits.

From there he was given a calculus book, the title of which I cannot remember. I never got that far.

I suspect you have to at least follow the same path to have the same intuition.

I sometimes get the feeling that we seem to have taken a huge step backwards in math books over the past 50 years. Back when I was in college and studying multivariate calculus I happened to find a small, ~100 page, book called something like "Introduction to Multivariate Calculus" from the 50s in a used book store. This tiny books not only covered basically the whole curriculum of my course, but did it in much greater clarity then the 500+ page that was our textbook. I can basically thank that book for me passing that course. I find on the whole that especially introductory mathbooks have gotten harder to follow and less clear (and a lot longer) over the past few decades.
Completely agree.

I've taken the liberty of taking a quick snap of a random page in "Arithmetic for the Practical Man" to include below for those poor people poisoned by modern textbooks: (926KiB)

I see horrible modern behemoths of over a 1000 pages that leave you dazed, confused and full of facts but nowhere to go with them. EE textbooks are even worse on this front than your average mathematics text book. I've seen one proudly promoting over 1500 pages and 1000 illustrations, but doesn't even get as far as an opamp or discuss anything at system level.

bought the series as well, love it. It's my daughter's favorite math series. One interesting thing I noticed in this regard is textbooks from the 30-60s have way more textual descriptions. They seem to spend more time looking at the problem or concept in a literary way and that might have helped to build a better understanding for the student.
Nice, simple, intuitive proofs. Very cool.
I think it's a bit misleading to say he learned math from those books. He got his start there, but surely the bulk of his mathematical knowledge was more advanced. However, it's quite possible that he retained the attitude from those early books. It seems to me though that he already had that attitude prior to reading the "practical man" books, and it is more that they particularly resonated with him because of it.
That's the most egotistical thing I've ever read. It's pretty typical of physicists though. Whenever you see a 'scientist' on television pontificating about things they don't really understand (from psychology to economics to theology to whatever happens to be relevant) they're usually physicists.
But there can't be, because mathematics is abstract and most of its fields have no intuitive physical analogies to back them up.
I don't think it's the intuition. I think it's the part where people are explicitly and implicitly taught to avoid metaphors, since they are considered bad analogues and "window dressing on top of objective literal truths". The sad part it, Lakoff and Johnson already provided a good counter-argument that thesis in the eighties with their landmark 'Metaphors We Live By,' suggesting that metaphors are the main way humans make sense of the world, almost as if they are the fundamental intuition you refer to. Since then then the proof for this case has only been piling up.

Especially in the field of machine learning we're finding more hard evidence that metaphor are not decoration, but fundamental parts of how to transfer information. Using rich metaphors to pass on implicit information between teacher and student is known as "privileged information":

> When Vladimir Vapnik teaches his computers to recognize handwriting, he [he harnesses] the power of “privileged information.” Passed from student to teacher, parent to child, or colleague to colleague, privileged information encodes knowledge derived from experience. That is what Vapnik was after when he asked Natalia Pavlovich, a professor of Russian poetry, to write poems describing the numbers 5 and 8, for consumption by his learning algorithms. (...) [After coming up with a simple way to "quantify" the poetry], Vapnik’s computer was able to recognize handwritten numbers with far less training than is conventionally required. A learning process that might have required 100,000 samples might now require only 300. The speedup was also independent of the style of the poetry used.

Now, of course, knowing how to come up with a good metaphor is a skill in itself, and bad metaphors do lead people astray. But they do so precisely because they are so good at transferring information - wrong information, in the case of bad metaphors.

Check out Linear Algebra: Foundations to Frontiers (, it's being offered self paced by edx, and you can download the lecture notes for free at When I took the course two years ago I liked how the lecturer will discuss basic concepts in the lectures but also gives additional material related to state of the art research being done in the field.

There also used to be a course called Coding the Matrix, I'm not sure if it is still being offered online. The lecture notes form a book of the same title, which is available for less than 10$ (Kindle).

Sep 11, 2016 · rz2k on Machine Learning in a Year
"Linear Algebra from Foundations to Frontiers" on EdX[1] is an introductory course in linear algebra with a manageable learning curve. There are many calculus courses on Coursera and EdX.


I have worked a bit with Strang and I love the intiution that his lectures provide but I did not finish the full course :)

I initially wanted to suggest Linear Algebra from Foundation to Frontiers [1] as it had an accompanying book [2] from Edx, but I had no experience in that, so checked up a bit online and outside their course website, they had not so good reviews [3]. So I am providing the links for you to decide yourself. Another answer on quora [4] suggests Coding the Matrix from Coursera, which is supposed to be rigorous in algorithms with excellent assignments in real world applications. So may be that is your cup of tea.





shoutout to LAFF as well



might be good for you if you're motivated by a deadline, graded progress, some Matlab/Octave implementation, and certificate of accomplishment, vs. self-paced Strang (AFAIK).

Had no clue about this course! Thanks a lot for the recommendation!

LAFF is great

I agree. As far as calculus goes, I am more enamored with books like Spivak's ( that take a proof-centric approach to teach calculus from first principles.

Incidentally, for those who want to learn linear algebra for CS in a mooc setting there are 3 classes running at this very moment: (from UT Austin) (from Davidson) (from Brown)

The first 2 use matlab (and come with a free subscription to it for 6 months or so), the last python. One interesting part of the UT Austin class is that it teaches you an induction-tinged method for dealing with matrices that let you auto-generate code for manipulating them: .

And of course there are Strang's lectures too, but those are sufficiently linked to elsewhere.

My calc I course in university was applied calculus without a text. I had to go back and redo single variable by reading Spivak (and Polya's How to Solve It) to figure out the proofs in Concrete Math by D. Knuth, et. al.
Learn linear algebra while writing a linear algebra library using the latest techniques (starts at the end of this month): I think there's also a non-mooc version at

Unfortunately, I had to give up on the course last year because my math background is even more limited than yours (you'll need to know how to construct proofs). So time for me to learn calc, I guess. :)

I don't believe Calculus is a prerequisite to Linear Algebra. Constructing proofs is something that's used in all branches of math as far as I know. If proofs are the only thing stopping you then I'd recommend just trying your hand at a few to get the hang of them. I can't imagine they'd be taught in a Calculus course either.
Oh, I realize Calculus isn't a necessary prereq to linear algebra. But my impression is that people normally learn proofs alongside Calc than learn proofs alongside Linear Algebra. To be honest, I was writing more out of a desire to be humorous than accurate; sorry!

Anyways, I've picked up "The Haskell Road to Logic, Maths, and Programming," so hopefully I can learn proofs alongside programming and logic, which is probably the best route for me at this time.

Another upcoming edx course that might be of interest to people on here:

    Linear Algebra - Foundations to Frontiers

    Learn the theory of linear algebra hand-in-hand with the practice of software library development.
(can't post a thread of its own, as was submitted too recently)
I'm very much lacking in math education and this seems like it would be a great approach for learning linear algebra. Do you know how this compares with "Coding the Matrix: Linear Algebra through Applications to Computer Science"? The two approaches sound very similar.
I signed up for "Coding the Matrix" and didn't finish. Before that, the last math I had studied was Calc 1 during my freshman year of college, over a decade ago. My impression was that it seemed like a good class and a good professor, but it moved quite quickly and was hard to follow for people lacking a background in higher math. I did get very comfortable with writing comprehensions in Python, though.

I still want to learn Linear Algebra, myself, so if anyone has suggestions, please post them.

You could buy the book, which isn't that expensive and covers everything from the video's as far as I can tell.

That allows you to study it more at your own pace. I'm taking the Visualizing Algebra course from Udacity, after working halfway through Coding the Matrix, because I found my algebra skills to be lacking.

Too bad its no longer possible to get a certificate on Coursera though, now I have to wait until the next iteration.

Thank you for this feedback. You have most likely saved me a lot of angst. :)
Hit or whatever for "Strang Linear Algebra" and you'll see the MIT OCW videos for free. The quality is, um, very turn of the century, but you're watching to learn, not critique video codecs.

If you want to spend money, from my bookshelves:

Strang (the guy in the videos above) knows one or two things about Linear Algebra. His textbook is legendary. Bring lots of $$$, like three figures.

"The Manga guide to Linear Algebra" Yes, that is exactly what it sounds like. I think if you have to start somewhere, maybe this is it. Cheap. $

"Matrices for Engineers" by Kraus. There's about 50 textbooks along the lines of linear algebra matrices for engineers programmers using $math_application or $calculator or $chicken_entrails and similar title permutations. You'd think every engineering program in the nation is required to use a different text. This particular text was pretty good. If I recall correctly, reasonably priced $$.

I would suggest reading them in the order of the comic book, the engineer book (or any of the dozens of equivalent college texts), and Strang. Coincidentally thats also order of price.

At one time I understood everything in the comic book and the engineer book. That was a long time ago. Strang mystified me in parts. So I'm not going to pretend to have THE perfect answer. It is entirely likely in the last decade someone has written the Uber text to replace them all. Probably a new edition of Strang is out by now.

I would estimate the effort required to be about one programming language.

Best of luck to you.

IMHO the uber text for intro LA is Axler's.
Dec 05, 2013 · 3 points, 0 comments · submitted by jjhageman
Oct 06, 2013 · 3 points, 0 comments · submitted by ColinWright
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