Linear Algebra Done Wrong (2004)

https://www.math.brown.edu/streil/papers/LADW/LADW.html

by sebg • 2 years ago

75 59 2 years ago

math.brown.edu

dang• 2 years ago

Linear Algebra Done Wrong [pdf] - https://news.ycombinator.com/item?id=999494 - Dec 2009 (12 comments)

optbuild• 2 years ago

After having my first Linear Algebra course I came across the online course called Linear Dynamical Systems by Prof Stephen Boyd of Convex Optimisation fame.

Every lecture was so eye opening. I couldn't believe that linear algebra could be taught in such a context with such a variety of application domains.

The lectures are all available online with the assignments : https://ee263.stanford.edu/archive/

nextos• 2 years ago

Boyd has published a related book which is a bit more elementary but still great: https://web.stanford.edu/~boyd/vmls/

abhgh• 2 years ago

I hadn't known about this, so thank you! Steven Brunton also has a book that covers dynamical systems (among other things) which is linear algebra adjacent, and the expositions are great [1].

[1] https://www.amazon.com/Data-Driven-Science-Engineering-Learn...

behnamoh• 2 years ago

We seriously need better terminology, notation, and pedagogy when it comes to linear algebra. In 2024, such old-style text books just don't cut it anymore.

AnimalMuppet• 2 years ago

What, specifically, do you think should be done better? What style of pedagogy do you think would be more appropriate? What notation? What terminology?

Apart from "argument by year number", what's actually wrong with the book?

Jtsummers2• 2 years ago

1. What's wrong with this specific book other than that it hasn't been updated in 7 years?

2. What are examples of better terminology, notation, or pedagogy that are improvements over this book? (Doesn't need to be total, could retain the same terminology and notation but just have better pedagogy for example)

dunefox2• 2 years ago

I think interactive linear algebra could be superior to a pure textbook. So, a textbook with interactive visualisations that show a concept and allow manipulation. Something like this maybe: https://textbooks.math.gatech.edu/ila/

andrewla2• 2 years ago

As far as I know, this notation and terminology is still very standard.

In some ways I like the idea of replacing traditional indexing with something more like the Einstein summation notation, and moving away from the arbitrary feeling that you get around how matrix multiplication is structured, and provide an immediate route into tensor theory.

But so many things that are very important in linear algebra, like matrix inverses, are awkward to express in this notation.

paulpauper• 2 years ago

Why does linear algebra elicit so much confusion despite being an elementary topic. I think part of the problem is that the connection between matrices and matrix operations and their applications is not explained well enough. Linear algebra is typically one's first foray into abstract mathematics, and if taught poorly sets one up for failure down the road.

rguzman• 2 years ago

one issue that comes to mind is the nature of the determinant. when one considers the determinant defined by the recursive definition, it seems like a highly contrived object that is difficult to work with (as it is from that definition!). avoiding that confusion requires that a lot more scaffolding be built (ala Axler in the "Done Right" book). either way you have some work: either to untangle the meaning of the weird determinant or get to the place where you can understand the determinant as the product of the eigenvalues.

cooljoseph• 2 years ago

Do you know why introductory textbooks don't define the determinant in terms of the exterior product? This is how some "real" mathematicians I've talked to define it. It also is more intuitive (in my opinion) to define determinants as "signed volumes" than some sum of products multiplied by signs of cycles.

The product of eigenvalues definition is also somewhat intuitive to me ("How much does the matrix scale vectors in each direction? Now multiply those numbers together."), but it's harder to motivate the fact that adding rows together doesn't change the determinant, which is kind of important to actually computing the determinant.

QuercusMax• 2 years ago

I took LinAlg the same semester I took Computer Graphics, back in 2001. The first half of LinAlg was all about solving equations, transformation matrices and that sort of stuff, and the first half CG was all 2D drawing (Bresenham and such). Second half of CG was all about OpenGL, and I was able to apply all the stuff I'd just learned in LinAlg.

The second half of LinAlg was a bunch of stuff about Eigenthis and Eigenthat, with no particular explanation as to what you'd use it for or why you would care. I pass the class with an A and was even recommended by the instructor to be work as a tutor for other students, but 23 years later I couldn't tell you the use of any of that stuff.

T-matrices, though - I've used that stuff my whole career, working in medical imaging and related fields. I still couldn't tell you what an Eigenvalue or Eigenvector is useful for, as I never had any reason to apply that stuff.

It's not particularly abstract if you can explain why and how this stuff is used - and I know it's heavily used in many fields - just not those that I've worked in.

will-burner2• 2 years ago

Yeah, eigenvectors and eigenvalues are abstract and best understood in terms of the linear mapping between vector spaces and change of basis, but change of basis for a vector space and writing a matrix in a new basis gets really complicated. Many people get lost there.

FabHK• 2 years ago

Eigenvectors characterise a linear transformation by answering a simple question: What lines map to themselves after the transformation?

Say for example you rotate something in 3D. The rotation axis remains unchanged. That's an eigenvector. Or say you mirror something in 3D, then all the lines lying in the mirror plane remain unchanged (all eigenvectors with eigenvalue 1), and the line orthogonal to the mirror plane remains unchanged - or rather, flipped, so it's an eigenvector with eigenvalue -1.

will-burner2• 2 years ago

It's hard to make the connection between matrices and a linear mapping between vector spaces. A lot of the time students in a linear algebra classes do not have a great grasp of what a function is, so when talking about vector spaces and linear maps between them they're lost. Then connecting that to a matrix is hopeless.

hintymad2• 2 years ago

Why does linear algebra elicit so much confusion despite being an elementary topic

I'm not sure that linear algebra is an elementary topic, if the word "elementary" is as in math of elementary school. That said, I don't think there's much confusion about linear algebra either. It's more likely that linear algebra is substantially more abstract than high-school algebra, so many students have a hard time deeply understanding it, just like many students already bail out on high-school math. When I was a TA in college, I also observed that many students were not prepared for the fact that college-level math is fundamentally different from high-school ones. College-level maths has far less time for students to grasp the concepts through sheer brute-force practice. College-level maths requires students to focus on intuitive understanding of they key concepts so the students won't get bogged down by hundreds and hundreds of concepts or corollaries or theorems. Of course, high-school maths requires intuitive understanding too, but because high-school maths is so simple that many students get the intuition naturally so they are not aware of how important such intuitive understanding is.

This book used to help my students build intuitions: https://www.amazon.com/Algebra-Through-Geometry-Undergraduat.... It starts with 2D geometry to teach linear algebra, and then moves to 3D, and then 4D. The author also chooses to use calculations and constructions to prove most of the theorems instead of more abstract or algebraic approach.

jwells89• 2 years ago

Thinking back to high school and college, the biggest issue with I had math in both is how it's frequently taught in absence of practical examples. Speaking personally (though I believe there are many who feel similarly), there's a need to see real world examples to develop a true grasp on new concepts in any reasonable amount of time, for reasons related to both motivation and intuition.

mamonster• 2 years ago

Because it's taught in the 1st year of most math courses and most people don't take it seriously and think that it's "easy 1st year math" when its actually the base for a whole lot of maths down the line.

I've actually stopped asking questions about Cayley-Hamilton and Jordan form (both covered in year 1 of any decent BS math course in Europe) when I used to do inteviews for trading/quant positions because so many people failed.

xanderlewis4• 2 years ago

You can probably reduce it down to a single definition. As soon as matrix multiplication is introduced without careful motivation, almost everyone is lost. And those who aren’t should be.

It’s mostly because, as Axler explicitly tries to address, linear algebra is a subject that can be viewed through at least two different lenses: the intuitive, (possibly higher-dimensional) geometric lens, or the algorithmic and numerical lens. Of course they are equivalent, but almost every course in linear algebra teaches the second and barely even touches on the first. It’s like learning Euclid’s algorithm before you’ve learnt to count. No wonder everyone’s so confused.

myworkinisgood2• 2 years ago

Linear algebra is definitely not elementary. There are layers and layers of linear algebra, and depending on the audience, you have to select which topics to emphasize.

generationP3• 2 years ago

FYI, there is a newer version at https://sites.google.com/a/brown.edu/sergei-treil-homepage/l... . Google seems to never have learnt about this site, which is weird as it is on a Google server.

godelski7• 2 years ago

Something seems to be seriously wrong with google. By "something" I mean "a lot".

I know there are Google engineers here, possibly search. Do you guys not also feel it? I mean you dogfood, right? I know it is a hard problem but boy has it become harder to use, especially in the last 6 months.

akira25012• 2 years ago

Like.. the very fact I typed "javascript" or "css" or "html" means you probably don't need to apply your "did you mean" logic to my searches. If I'm searching for something highly technical, I absolutely DO NOT NEED your help, Google.

junar• 2 years ago

"Linear Algebra Done Wrong site:sites.google.com" finds the page you linked.

https://www.math.brown.edu/streil/index.html redirects to https://sites.google.com/a/brown.edu/sergei-treil-homepage/h...

But OP's link, https://www.math.brown.edu/streil/papers/LADW/LADW.html does not redirect. So it seems like a problem that the site owner can fix.

godelski7• 2 years ago

  > "Linear Algebra Done Wrong site:sites.google.com" finds the page you linked.

I think you have the wrong framing. It isn't that you can't create a p̶r̶o̶m̶p̶t̶ query that yields the correct result, it is the ability to do this without knowing the answer a priori.

As in, if you didn't know a second newer version existed, would you find it? Certainly the new version should prioritize the older version.

FWIW, my query "Linear Algebra Done Wrong" yields the newer version (2017), but I've just interpreted OP's point as being illustrative and helpful to readers (and it looks like the link now does point towards the new version. Possibly this made my query work)

generationP3• 2 years ago

You can also get to the new site from one of the hits further down ( https://textbooks.aimath.org/textbooks/approved-textbooks/tr... ) if you pay attention. But it's far from the transparency I'd have expected googling an openly available and multiply linked website.

aquafox• 2 years ago

For a really intuitive introduction to linear algebra, I highly recommend Gilbert Strang's "Linear Algebra and Learning from Data": https://math.mit.edu/~gs/learningfromdata/

godelski7• 2 years ago

2011 lectures: https://www.youtube.com/playlist?list=PL221E2BBF13BECF6C

Also useful: https://www.youtube.com/playlist?list=PLUl4u3cNGP61iQEFiWLE2...

munchler2• 2 years ago

Why should anyone read this book if it presents the subject in a wrong way? What is particularly done “wrong” in the book?

Does he ever answer this question? It's posed at the start of the book, but then immediately ignored.

Syzygies2• 2 years ago

The title is a jab at and a response to "Linear Algebra Done Right" (https://linear.axler.net) by Sheldon Axler.

Sheldon's book is more suitable for a second course in linear algebra. It's an excellent book, that my more advanced students have enjoyed.

Sheldon, however, approaches the subject with some rigid dogma: Determinants are evil. His view is they're not practical to compute at scale. It can be good to learn theory with one's determinant hand tied behind one's back, just as it can be good to learn "constructive" mathematics, but this should not be one's only approach.

I used to attend an academic sponsor's day at the math institute MSRI (now SLMath). Afternoons reached to keep us busy. One year Sheldon's wife demonstrated some academic teaching software, and my group was tasked to solve a linear algebra problem that Sheldon had provided. Unaware that everyone was watching us on the big screen, I solved it instantly, using determinants. I still don't understand Sheldon's longer solution.

munchler2• 2 years ago

This is very informative. Thank you for the explanation.

btilly• 2 years ago

His view is not just that they are not practical to compute at scale. (They actually are practical to compute. Just not with the formula. Use row reduction instead.)

His view is that they are presented as an arbitrary piece of magic that people really don't understand. And, exactly because they don't really understand the determinant, they wind up not properly understanding fundamental concepts in linear algebra. Such as Jordan normal form.

If your goal is actually to properly understand linear algebra, he's absolutely right. If your goal is to use linear algebra for a wide variety of practical purposes, though, determinants are often essential.

beryilma• 2 years ago

... and another beauty of a math book written in LaTeX is that you can fit a 286-page book into just 1.3 MB. Good luck doing that in MS Word.

Linear Algebra Done Wrong (2004)

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