How should I learn Linear Algebra?
May 27, 2022 4:46 PM Subscribe
I'm a mathematically-minded person, but informally; I was great with math in high school and so-so in college, didn't deal well with calculus, love recreational mathematics on my own terms, and do best when I have a good conceptual basis for any bit of mathematical theory. (I do great with planar geometry. I do okaaay with symbolic algebra.) But I never learned linear algebra and matrices, despite them applying to a *lot* of things I'm otherwise interested in. Where's a good, accessible, concept-driven way to get familiar with lin alg territory?
Response by poster: I'll take a peek in the spirit of investigation, but based on my familiarity with HN I would say I am to some extent interested in precisely the sort of answers they wouldn't typically put out.
posted by cortex at 5:05 PM on May 27, 2022 [2 favorites]
posted by cortex at 5:05 PM on May 27, 2022 [2 favorites]
YouTube search for 'linear algebra' brings up many, but here's one that's a course introduction entitled "What's the big idea with linear algebra" which looks like it might be a place to start.
I remember not being as good with matrices as with algebra, but better with either than with planar geometry.
I am curious: in my experience long ago it seemed my peers, except really exceptional cases, would either get As in Algebra and Algebra II with a B or B+ in Geometry, or Bs in the Algebra classes and an A in Geometry. Has that been your experience?
posted by TimHare at 5:20 PM on May 27, 2022
I remember not being as good with matrices as with algebra, but better with either than with planar geometry.
I am curious: in my experience long ago it seemed my peers, except really exceptional cases, would either get As in Algebra and Algebra II with a B or B+ in Geometry, or Bs in the Algebra classes and an A in Geometry. Has that been your experience?
posted by TimHare at 5:20 PM on May 27, 2022
Can you grab Unity, and start messing with GameObjects parented to other GameObjects? Each GameObject has a transform, and even though you have to (IIRC) modify them by setting the translation, rotation, and scale as 3 separate properties, you can read them out via gameObject.transform and see what impact the changes have?
I mostly learned by writing my own toy GL2 immediate-mode engine for a project long ago; suspect doing the same thing in Vulkan or etc might be a bit intense now, but if you go through some of the Vulkan tutorials you'll end up multiplying at least a few vectors and matrices, but that may be too close to "stuff recommended on HN"...
posted by Alterscape at 6:00 PM on May 27, 2022
I mostly learned by writing my own toy GL2 immediate-mode engine for a project long ago; suspect doing the same thing in Vulkan or etc might be a bit intense now, but if you go through some of the Vulkan tutorials you'll end up multiplying at least a few vectors and matrices, but that may be too close to "stuff recommended on HN"...
posted by Alterscape at 6:00 PM on May 27, 2022
Response by poster: I like the *idea* of fucking around conceptually in a 3D environment like Unity and getting my footing there—my first grudging exposure to matrices was in fact the 3D programming OpenGL-centric course I took in college (which in retrospect understandably had a Lin Alg prerequisite I bargained my way out of)—but in practice I've bounced hard of the whole Unity paradigm in the past and I think something more focused on specifically an understanding of matrices and transformation concepts in their own right outside of a dev environment would be more approachable for me. I get very frustrated very easily when Other Logistical Stuff impinges on my engagement with a core idea.
posted by cortex at 6:08 PM on May 27, 2022
posted by cortex at 6:08 PM on May 27, 2022
Came for 3Blue1Brown.
posted by zengargoyle at 6:34 PM on May 27, 2022
posted by zengargoyle at 6:34 PM on May 27, 2022
Response by poster: 3B1B is great and I’m a fan; I’ve watched at least a couple things from them on the subject.
I think if I had framed this more effectively up front I could have said basically: that accessible sensibility but long-form and a liiiittle more formal. Something that goes farther, and a little more deliberately, than a condensed half hour youtube pop-math treatment with its necessary constraints, if that makes sense.
posted by cortex at 7:11 PM on May 27, 2022
I think if I had framed this more effectively up front I could have said basically: that accessible sensibility but long-form and a liiiittle more formal. Something that goes farther, and a little more deliberately, than a condensed half hour youtube pop-math treatment with its necessary constraints, if that makes sense.
posted by cortex at 7:11 PM on May 27, 2022
I think you'd like a classic advanced undergrad textbook.
I don't think you want to hassle with doing lots of row reduction etc by hand, but I do think you want a good grounding in linear (in)dependence, the idea of solving linear systems of equations, and the geometry of things like orthonormal bases and matrix representations of linear transformations.
I love how linear algebra builds bridges to geometrical modeling of stuff, and the usefulness for understanding linear dynamical systems. Ordinary differential equations also make a lot more sense if you approach them from a viewpoint of linear algebra and linear dynamics. You don't necessarily need much calc to get access to and value from these ideas, so long as you are ok with what differentiation and integration are as fundamental concepts (no need for detailed problem solving)
I could maybe say more if you describe more of your goals.
On balance, try this book, or this one. You can maybe use inter library loan or grey market methods to try before you buy.
posted by SaltySalticid at 8:18 PM on May 27, 2022 [4 favorites]
I don't think you want to hassle with doing lots of row reduction etc by hand, but I do think you want a good grounding in linear (in)dependence, the idea of solving linear systems of equations, and the geometry of things like orthonormal bases and matrix representations of linear transformations.
I love how linear algebra builds bridges to geometrical modeling of stuff, and the usefulness for understanding linear dynamical systems. Ordinary differential equations also make a lot more sense if you approach them from a viewpoint of linear algebra and linear dynamics. You don't necessarily need much calc to get access to and value from these ideas, so long as you are ok with what differentiation and integration are as fundamental concepts (no need for detailed problem solving)
I could maybe say more if you describe more of your goals.
On balance, try this book, or this one. You can maybe use inter library loan or grey market methods to try before you buy.
posted by SaltySalticid at 8:18 PM on May 27, 2022 [4 favorites]
Lang’s Linear Algebra has gone through several editions so the old ones are cheaper.
posted by clew at 8:33 PM on May 27, 2022 [2 favorites]
posted by clew at 8:33 PM on May 27, 2022 [2 favorites]
Stanford CS229 has a very good informal (i.e. CS/physicist friendly) but terse reference PDF that I keep bookmarked. It's not a teaching document, but I learn (or remember) something whenever I look at it.
Otherwise, linear algebra is literally the perfect subject for teaching with video, and the Khan Academy course is pretty good.
posted by caek at 8:41 PM on May 27, 2022 [2 favorites]
Otherwise, linear algebra is literally the perfect subject for teaching with video, and the Khan Academy course is pretty good.
posted by caek at 8:41 PM on May 27, 2022 [2 favorites]
To look at this another way, you might go for an entry through quantum mechanics or another -applied- linear algebra approach. It’s so just much dancing around until you have a reason to use it to solve a problem.
posted by janell at 9:34 PM on May 27, 2022 [2 favorites]
posted by janell at 9:34 PM on May 27, 2022 [2 favorites]
Do you have any issues with trypophobia? I got an A in my linear algebra course in college, but it was a truly undeserved gift, and I never did really grasp it.
And matrices were the worst! I just did not like manipulating them, but it took a MetaFilter thread and some introspection for me to connect it to trypophobia, which I think also explains my aversion to buttons. I remember a dish full of buttons on my mother's dresser for which I felt strong revulsion from an early age.
posted by jamjam at 11:21 PM on May 27, 2022 [1 favorite]
And matrices were the worst! I just did not like manipulating them, but it took a MetaFilter thread and some introspection for me to connect it to trypophobia, which I think also explains my aversion to buttons. I remember a dish full of buttons on my mother's dresser for which I felt strong revulsion from an early age.
posted by jamjam at 11:21 PM on May 27, 2022 [1 favorite]
To look at this another way, you might go for an entry through quantum mechanics or another -applied- linear algebra approach.
Strongly agree. I don’t think physicists and engineers really learn linear algebra in math classes—they learn it in an applied context. And there are any number of texts that teach the linear algebra quite formally alongside the application, with good mathematical rigor. So is there a topic in physics or engineering that you’re interested in? I’d probably recommend classical mechanics over quantum if you’re mostly interested in the math.
posted by mr_roboto at 12:02 AM on May 28, 2022
Strongly agree. I don’t think physicists and engineers really learn linear algebra in math classes—they learn it in an applied context. And there are any number of texts that teach the linear algebra quite formally alongside the application, with good mathematical rigor. So is there a topic in physics or engineering that you’re interested in? I’d probably recommend classical mechanics over quantum if you’re mostly interested in the math.
posted by mr_roboto at 12:02 AM on May 28, 2022
Programming, and specifically, computer graphics, would be my suggestion for you. There are also many interesting linear algebra type problems that can arise in other types of programming, too. I can expound / help look for references if this sounds appealing.
posted by dbx at 4:50 AM on May 28, 2022
posted by dbx at 4:50 AM on May 28, 2022
I could have written a variation of this question. I somehow managed to pass a linear algebra class in college without really understanding what a vector is, and so I've recently been trying to go back to basics and fill in the gaps
They're little more traditionally formal, but I really like the MIT OCW videos from Gilbert Strang. I find his manner comforting, and I think his explanations are quite good. Even if self study of the whole course is not your style, I think just this first lecture stands alone and personally switched my mindset from "linear algebra is something I need to understand these other things" to "oh neat linear algebra is actually interesting to me on its own"
posted by okonomichiyaki at 4:59 AM on May 28, 2022 [3 favorites]
They're little more traditionally formal, but I really like the MIT OCW videos from Gilbert Strang. I find his manner comforting, and I think his explanations are quite good. Even if self study of the whole course is not your style, I think just this first lecture stands alone and personally switched my mindset from "linear algebra is something I need to understand these other things" to "oh neat linear algebra is actually interesting to me on its own"
posted by okonomichiyaki at 4:59 AM on May 28, 2022 [3 favorites]
okonomichiyaki beat me to the Strang video recommendation. He goes slow, but if you rush through math then you get to a point where your tools aren't good enough to go further, and you asked for long-form anyway. He has a style of pretending to discover the material as he goes that may seem hokey at first, but it's really effective.
posted by dfan at 5:23 AM on May 28, 2022
posted by dfan at 5:23 AM on May 28, 2022
Came to suggest Professor Strang even though I'm only a couple sessions in. He does have an wry in-joky style that I think is great but it is for pretty motivated undergrads. Definitely watch the 3B1B videos for intuition.
There seems to be two "tracks" in the LA math world, the pragmatic and theoretical. And rather amusingly two respected texts are titled "Linear Algebra Done Right" (the strongly theoretical based approach) and another with the inverted title "Linear Algebra Done Wrong" (focused on the how to, which is still heavy math for most folks). So unless you're looking at moving into more deeply abstract areas of math (topology, abstract algebra (which ok messes with my head as algebra is pretty darned abstract already:-), group theory, or heavy proof oriented topics) orienting towards the pragmatic (hate to say "how to" but..) may be slight more satisfying/useful.
posted by sammyo at 6:48 AM on May 28, 2022 [1 favorite]
There seems to be two "tracks" in the LA math world, the pragmatic and theoretical. And rather amusingly two respected texts are titled "Linear Algebra Done Right" (the strongly theoretical based approach) and another with the inverted title "Linear Algebra Done Wrong" (focused on the how to, which is still heavy math for most folks). So unless you're looking at moving into more deeply abstract areas of math (topology, abstract algebra (which ok messes with my head as algebra is pretty darned abstract already:-), group theory, or heavy proof oriented topics) orienting towards the pragmatic (hate to say "how to" but..) may be slight more satisfying/useful.
posted by sammyo at 6:48 AM on May 28, 2022 [1 favorite]
If you want a decent textbook, I’m a fan of David Lay’s Linear Algebra—it’s what I teach from, although the new editions are overly spendy. It’s in at least the 5th edition, so you should be able to find the third used. (I like that one because it has a nice chapter on convex geometry.) I like the way it defines matrix multiplication, and it’s got really good true-false questions. I don’t know if you can find an instructor’s edition used. (But that thriftbooks edition is cheap enough you could just buy it as a supplement to whatever else you find.)
If you want quizzes or tests to look at to test your knowledge, let me know, I’d be happy to share some old stuff. Or lecture notes. I like linear algebra.
posted by leahwrenn at 6:50 AM on May 28, 2022 [3 favorites]
If you want quizzes or tests to look at to test your knowledge, let me know, I’d be happy to share some old stuff. Or lecture notes. I like linear algebra.
posted by leahwrenn at 6:50 AM on May 28, 2022 [3 favorites]
If you like planar geometry, and kind of pencil-and-papering your way through puzzles like that, then you might approach linear algebra by messing around with problems in the same way. Grokking linear algebra initially can just mean developing an intuition about how components of vectors and matrices relate -- almost exactly the same way that you would develop an intuition for triangles and circles and right angles and areas by futzing around with geometry problems.
If that sounds like your kinda thing, then I would recommend books and courses that feel like David Lay's book in terms of style, and to focus on working out problems and doing lots of concrete examples. The great thing about linear algebra at this scope is that you can always guess-and-check, and verify results by literally writing out little 2x2 example matrices, or -- even better, drawing out the vectors and other geometric objects that appear and seeing how they transform with different operations. Like, oh this is what a rotation matrix feels like because I can very explicitly watch it transform my little 2d example vector as I plug in different values and notice that the length of that vector never changes because ..., then zooming out to view it in terms of basis changes where the columns of the matrix mean ..., why it's not a symmetric matrix, linearity when acting on vectors, etc. And definitely make liberal use of Wolfram Alpha.
It's not like you need to manipulate anything beyond 2d (or very occasionally 3d) objects to get the idea of a theorem; the point is not to be able to calculate determinants of very big matrices -- because who the hell does that by hand -- but to use a lot of small examples to develop an intuition for the more abstract ideas and constructions, and to feel like in principle you could go back to writing out all of the numbers in big tables and rows and that there was a way to turn the crank and eventually calculate what you wanted.
I'm not sure about what level of familiarity you already have with this kind of thing so I might not be speaking to your actual level. If you feel like all that is child's play, then any of the other more structured resources listed above would be great -- Dettman, Lang, etc. Pick one and go with it. But if any of those ever feel over your head, or if small mechanical steps like 2d matrix-vector multiplication feel clunky, then I would focus on solving explicit problems to develop that concrete intuition and fluidity first. Unless you really get a lot out of abstract expositions then there is no substitute for a lot of hands-in-the-dirt problem solving.
P.S. if you're interested in quantum physics and/or the question of "how did matrix rules even happen" then you may have a good time looking into this.
posted by miniraptor at 9:36 AM on May 28, 2022 [1 favorite]
If that sounds like your kinda thing, then I would recommend books and courses that feel like David Lay's book in terms of style, and to focus on working out problems and doing lots of concrete examples. The great thing about linear algebra at this scope is that you can always guess-and-check, and verify results by literally writing out little 2x2 example matrices, or -- even better, drawing out the vectors and other geometric objects that appear and seeing how they transform with different operations. Like, oh this is what a rotation matrix feels like because I can very explicitly watch it transform my little 2d example vector as I plug in different values and notice that the length of that vector never changes because ..., then zooming out to view it in terms of basis changes where the columns of the matrix mean ..., why it's not a symmetric matrix, linearity when acting on vectors, etc. And definitely make liberal use of Wolfram Alpha.
It's not like you need to manipulate anything beyond 2d (or very occasionally 3d) objects to get the idea of a theorem; the point is not to be able to calculate determinants of very big matrices -- because who the hell does that by hand -- but to use a lot of small examples to develop an intuition for the more abstract ideas and constructions, and to feel like in principle you could go back to writing out all of the numbers in big tables and rows and that there was a way to turn the crank and eventually calculate what you wanted.
I'm not sure about what level of familiarity you already have with this kind of thing so I might not be speaking to your actual level. If you feel like all that is child's play, then any of the other more structured resources listed above would be great -- Dettman, Lang, etc. Pick one and go with it. But if any of those ever feel over your head, or if small mechanical steps like 2d matrix-vector multiplication feel clunky, then I would focus on solving explicit problems to develop that concrete intuition and fluidity first. Unless you really get a lot out of abstract expositions then there is no substitute for a lot of hands-in-the-dirt problem solving.
P.S. if you're interested in quantum physics and/or the question of "how did matrix rules even happen" then you may have a good time looking into this.
posted by miniraptor at 9:36 AM on May 28, 2022 [1 favorite]
Cortex please update this thread with whatever you decide to go with and your findings. Very interesting.
posted by jouke at 1:17 PM on May 28, 2022
posted by jouke at 1:17 PM on May 28, 2022
The University of Waterloo's Open Math courseware.
posted by avocet at 2:52 PM on May 28, 2022 [1 favorite]
posted by avocet at 2:52 PM on May 28, 2022 [1 favorite]
Michael Penn's YouTube channel has videos that go along with his Linear Algebra course. His content is very good.
Edited to add: Penn has two channels. The Linear Algebra videos are on his Math Major channel.
posted by wittgenstein at 4:46 PM on May 28, 2022
Edited to add: Penn has two channels. The Linear Algebra videos are on his Math Major channel.
posted by wittgenstein at 4:46 PM on May 28, 2022
Also Paul's online math notes used to have a set for linear algebra. The ones for linear algebra are no longer posted on his site, but I hear that if you put "Paul's linear algebra notes" into a search engine, a pdf "should" pop up.
posted by oceano at 6:25 PM on May 31, 2022
posted by oceano at 6:25 PM on May 31, 2022
This thread is closed to new comments.
posted by bitslayer at 5:01 PM on May 27, 2022