[HN Gopher] Ask HN: Best beginner friendly linear algebra book? ___________________________________________________________________ Ask HN: Best beginner friendly linear algebra book? Hello all, the title really says it all. Hoping to find a linear algebra book that is friendly for visual learners. EDIT: thank you all for the great responses! Author : belfalas Score : 106 points Date : 2022-06-11 19:00 UTC (4 hours ago) | throwaway81523 wrote: | I liked Hirsch and Smale's old book called something like "Linear | algebra, differential equations, and dynamical systems". It is | now replaced by an expanded edition with a 3rd author added and a | longer title, that I expect is also good, though I haven't looked | at it. | | I don't know if the H&S book is beginner friendly, but what I | found good about it was studying linear algebra and differential | equations at the same time, i.e. treating them as closely related | topics rather than separate ones. So you could use your physical | intuition about (say) a harmonic oscillator (mass on a spring, | the archetypal second order ODE), then see how the 2nd order | equation can be separated into a system of first order ODE's, and | solved by finding matrix eigenvalues. | | That worked better for me than the abstract linear algebra | approach that was purely about vector spaces with nothing going | on in them. It showed real sensible motivations of linear | algebra. | jerDev wrote: | YouTube. Khan academy. There are so many people trying to make a | buck with a whiteboard online. Find one that you like ( gender, | nationality, accent, whatever works for you ) and then stick to | it | krosaen wrote: | Strang has a newer book aimed at being more approachable and | tying in a preview of deep learning and other modern topics. I | like it a lot. | | "Linear Algebra for Everyone" | | https://math.mit.edu/~gs/everyone/ | harshreality wrote: | Try Singh's _Linear Algebra: Step by Step_, along with youtube. | | Higher math tends to be abstract; you can't visualize higher- | dimensional linear algebra concepts directly. The standard | resources (Strang, Axler, etc) are worth the effort. | abxytg wrote: | What are you learning for? I'm in the industry learning for work | in medical image visualization. | ThatGeoGuy wrote: | "Linear and Geometric Algebra" by Alan Macdonald. | | It's definitely not the norm compared to many of the other | listings in this thread but it definitely gave me a better | understanding of many algebraic properties and helped build an | intuition around spaces, vectors, products, etc. | | It doesn't have a ton of graphics, to which you might snub your | nose at it (you mentioned visual learning), but the graphics it | does have are incredibly useful for building a geometric | understanding of what linear algebra concepts map to. The | subsection on quaternions and pseudoscalars is one of the best | descriptions of such in my experience. | nojito wrote: | Nothing holds a candle to https://web.stanford.edu/~boyd/vmls/ | | Applied learning is the best way to learn linear algebra. | saeranv wrote: | I wouldn't reccomend Strang's "Introduction to Linear Algebra" | textbook to a beginner. Strang has a very odd, dense way of | writing, often with references to material that has yet to be | introduced. I think this is a consequence of it's intended use as | an aid to his lectures, and can't really stand on its own. The | goodreads reviews on the textbook seems to share my opinion: | https://www.goodreads.com/book/show/179700.Introduction_to_L... | | I think it's great for an intermediate student, or someone who's | also watching Strang's lectures. | tptacek wrote: | I agree, but would recommend his video series to beginners. The | books themselves are less important than the exercises in the | book, which unfortunately sort of demand that you have the book | because they refer back to them. But the package of (exercises, | video lectures, book), in descending order of importance maybe, | I think is a worthy recommendation. Ultimately, the book is the | only part of that you actually have to "acquire", so it might | be ok that it doesn't stand on its own. | dragontamer wrote: | Why are you trying to learn linear algebra? | | This is highly important. Linear algebra is applicable to so many | fields, but learning linear algebra for say... Graphics | Programmers, is a completely different feel from learning linear | algebra for an Electrical Engineer Signals-and-systems engineer. | | Graphics programmers largely need to learn "how to use" | matricies. Emphasis on associative properties. Emphasis on non- | communitive operations. | | In contrast, Electrical Engineers / Signals-and-systems want to | learn linear-algebra as a stepping stone to differential | calculus. In this case, you're going to be focusing more on | eigen-values, spring-mass systems / resonant frequencies, | applicability to calculus and other tidbits (how linear algebra | relates to the Fourier Transform). | | ---------- | | The graphics programmer (probably) doesn't need to learn | eigenvalues. So any textbook written as "linear algebra for | graphics programmers" can safely skip over that. | | The electrical engineer however needs all of this other stuff as | "part" of the linear algebra class. | | I'm sure other fields (statistics, error-correction codes/galois | fields, abstract algebra, etc. etc.) have "their own ways" of | teaching linear algebra that is most applicable to them. | | Yes, "linear algebra" is broadly applicable. But instead of | trying to "learn all of it", you should instead focus on the | "bits of linear algebra that is most applicable to the problems | you face". That shrinks down the field, increases the | "pragmatism" of your studies. | | Later, when you're more familiar with "some bits" of linear | algebra, you can then take the next step of generalizing off of | your "seed knowledge". | | -------- | | I personally never was able to learn linear algebra from a linear | algebra book. | | Instead, I relearned linear algebra 4 or 5 times as the "basis" | of other maths I've learned. I learned it for differential | calculus. I relearned linear algebra for signals. I relearned | linear algebra for Galois fields/CRC-codes/Reed Solomon. I | relearned linear algebra for graphics. | | Yes, it seems inefficient, but I think my "focus" isn't strong | enough to just study it in the abstract. I needed to see the | "applicable" practice to encourage myself to learn. Besides, each | time you "relearn" linear algebra, its a lot faster than the last | time. | tzs wrote: | > I personally never was able to learn linear algebra from a | linear algebra book. | | > Instead, I relearned linear algebra 4 or 5 times as the | "basis" of other maths I've learned. I learned it for | differential calculus. I relearned linear algebra for signals. | I relearned linear algebra for Galois fields/CRC-codes/Reed | Solomon. I relearned linear algebra for graphics. | | If I were way better at websites and at advanced mathematics | than I actually am, I'd make a site for learning math in a top | down manner where you start with some result or application | that interests you and then are taught just enough more | elementary math to support that result or application. | | The site would have a list of results and applications, and for | each tell what math is necessary to understand it. You pick a | result or application that interests you, either because it is | interesting to you itself or because you see that it depends on | some more elementary math that you wish to learn. | | Once you pick, the site would show you a proof of the result or | development of the application, at a level that one would find | in a journal aimed at professionals in the relevant field. This | of course will most likely be largely incomprehensible at this | point. | | You can select any part of the proof or development and ask the | site for more information. There are two kinds of additional | information you can ask for. | | One is to ask for smaller steps. You use this when there is | some step A -> B where you are comfortable with A and B but | just don't see how it jumps from A to B. You understand what A | means, what B means, just not why A -> B. The site fills in the | intermediate steps. | | The other is to ask what something means. This is for when the | proof uses something you have not yet studies. For example if | the proof uses integration and you have not yet studied it | calculus that would be a great place to use a "what does this | mean?" request. The site would then give you a short | explanation of integration. | | A key feature of the site would be that this is all recursive. | If you use a "what does this mean?" request on an integral and | get the short explanation of integration, you could use | "smaller steps" requests and "what does this mean?" requests in | that explanation. | | Using "what does this mean?" requests recursively should let | you go all the way down to things that can be explained with | only high school algebra and precalculus. | | Note that if you've never studied anything past high school | algebra and precalculus and then use the site to learn | something like say an analytic proof of the prime number | theorem you will learn much elementary calculus but not all. | You will learn just what is needed for the prime number | theorem. | | But there would be other interesting theorems and applications | that use different parts of elementary calculus, so doing those | would fill in more of your elementary calculus. | | The site should have a planner that lets you pick areas of | undergraduate or masters level math that you would like to | learn and then shows you lists of interesting theorems and | applications it has that will cover those areas. | | I think this would be an interesting and effective way to | learn. At all points everything you are learning goes directly | toward supporting the top level proof you have chosen to learn, | and you have an idea of why it is useful because you are there | because you've already encountered something where you need it. | | I think that for many people this will provide better | motivation. In the conventional approach, where you do say a | whole class in calculus or abstract algebra, then do a more | advanced class that uses those results, and so on, a lot of | time you are learning stuff with no idea of why it is useful. | belfalas wrote: | Thank you, this is a great point! I am in the category of | someone who needs linear algebra in order to apply it for day- | to-day stuff, hands on not blue sky. Currently my primary use | case is image filtering but a bit down the line signal | processing will come up. | dragontamer wrote: | > Currently my primary use case is image filtering but a bit | down the line signal processing will come up. | | Image filtering _is_ signal processing, two-dimensional | signal processing to be precise. | | Traditionally, a college would take you through linear | algebra -> differential equations -> signals and systems, to | approach this subject. | | I found it easier to go through the reverse: start at | signals-and-systems (to see what you have to learn), then | work your way back down to linear algebra, and then work your | way back up to signals and systems. | | --------------- | | From a "signals and systems" point of view, your image | filtering functions are 99% going to just be a "kernel" | applied to an image. | | https://en.wikipedia.org/wiki/Kernel_(image_processing) | | IMO, its easier to start with a 1-dimensional version, where | you perform kernels upon sound and/or RADAR signals rather | than 2-dimensional images. | | https://en.wikipedia.org/wiki/Convolution#Visual_explanation | | You can see that the 1-dimensional version of the convolution | applied between (data x kernel) is extremely simple and | "obvious" to think about, given this GIF: https://upload.wiki | media.org/wikipedia/commons/6/6a/Convolut... | | Where blue-box is the original signal, and red-box is the | convolution-kernel, and the black-line is the output of blue | convolve with red. | | From there, you generalize the 1-dimensional convolution, | into a 2-dimensional convolution. To do so, you need to study | linear algebra and matricies. But now that you're "focused" | upon the convolution idea, as well as the idea of a "kernel", | everything should be "more obvious" to you as you go through | your studies. | | You can see that a "Matrix", in your specific field of study, | represents a kernel to a discrete system. The image you want | to manipulate is a 2-dimensional signal. A "matrix" is many | different things to many different mathematicians / | engineers. "Focusing" upon your particular application is key | to learning as quickly as possible. (You can generalize later | after you've mastered your particular field). | | Still, the study of signals / systems is a very generalized | and large field. Mechanical engineers study this, because it | turns out that an "impulse" that is "convoluted" with a | "kernel" is descriptive of how a speed-bump affects your | car's suspension system (!!!!). (EDIT: A youtube video | demonstrating the same math for earthquakes vs buildings: | https://www.youtube.com/watch?v=f1U4SAgy60c) | | So studying signals-and-systems is still a very abstract goal | of yours. It sounds like you need to focus upon the image- | processing portions of signals-and-systems. | | --------- | | IMO, you'll find that there's probably very little linear | algebra you actually need to learn for your particular path. | axegon_ wrote: | One which was posted here is an absolute masterpiece if you ask | me: | | https://news.ycombinator.com/item?id=24892907 | isaacimagine wrote: | I enjoyed _Linear Algebra Done Wrong_ [0], to be used in | combination with a more traditional textbook, like _Linear | Algebra and its Applications_ [1] (which has some good diagrams). | I 've already seen it mentioned, but I'd like to add that 3b1b's | _Essence of Linear Algebra_ [2] videos are well made and make for | a good supplementary resource early on. | | [0]: | https://www.math.brown.edu/streil/papers/LADW/LADW_2017-09-0... | | [1]: https://www.amazon.com/Linear-Algebra-Its- | Applications-5th/d... -- PDFs exist. | | [2]: | https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... | doppioandante wrote: | That is really a wonderful book which I perused when learning | linear algebra, maybe a bit on the mathy side for OP expecially | as he is asking a book for a "visual learner". Fortunately | linear algebra can be grasped intuitively in dimensions above 3 | even if it can't be visualized, but maybe I'm biased as it is | bread and butter for me now. | Syzygies wrote: | The title of _Linear Algebra Done Wrong_ is an unacknowledged | nod to Sheldon Axler 's _Linear Algebra Done Right_. I know | Sheldon; he believes it 's a crime to teach people | determinants. I teach people determinants. _Wrong_ features | determinants in Chapter 3. | | I was once part of an interactive learning software demo, where | Sheldon had provided the sample linear algebra problem. I | solved it in seconds using determinants. That really made my | day. | f0e4c2f7 wrote: | >I know Sheldon; he believes it's a crime to teach people | determinants. | | Any way to explain to a lay person why? | yCombLinks wrote: | Determinants are usually introduced in Linear algebra out | of the blue because you can't get to other important topics | in Linear Algebra without them. Calculating them is a | complex mess best left for a calculator. Sheldon teaches | Linear Algebra as a theoretical math course, along the | lines of learning Abstract Algebra. He approaches those | other important topics from a different direction entirely, | and determinants are just a trivial part of his book | because of the different approach. | mseri wrote: | The book is open access (https://linear.axler.net/), I am | going to quote directly the author in the preface: | | << all linear algebra books use determinants to prove that | every linear operator on a finite-dimensional complex | vector space has an eigenvalue. Determinants are difficult, | nonintuitive, and often defined without motivation. To | prove the theorem about existence of eigenvalues on complex | vector spaces, most books must define determinants, prove | that a linear map is not invertible if and only if its | determinant equals 0, and then define the characteristic | polynomial. This tortuous (torturous?) path gives students | little feeling for why eigenvalues exist. In contrast, the | simple determinant-free proofs presented here (for example, | see 5.21) offer more insight. Once determinants have been | banished to the end of the book, a new route opens to the | main goal of linear algebra-- understanding the structure | of linear operators.>> | | If you like mathematics, it is actually a pretty nice book. | na85 wrote: | I don't believe it's open access, or at least I see no | download link on that page. | hn_version_0023 wrote: | I'd ask a follow-up question of: what are the prerequisites for | being able to successfully complete any of these courses/books? | I've been thinking of doing something similar myself, and am 20 | years removed from daily math exercises. Thanks in advance! | tptacek wrote: | Algebra I and some trig, at least to get pretty deep into a | first college course syllabus and get enough exposure to see | where you want to go with it. That "10th grade" level of math, | for instance, is actually enough to get you pretty far into the | practical applications of linear algebra in cryptography, but | it's not enough to get you all the way to machine learning. | hn_version_0023 wrote: | Thank you kindly! Its always nice to learn you're more | prepared than you supposed! | wrycoder wrote: | I'd recommend Kahn Academy. They have a way of quickly | reviewing what you know. You ought to refresh any gaps in high | school math. Then take the Kahn courses in linear algebra. | | For more and deeper, see the other recommendations here. | photochemsyn wrote: | I'll recommend Linear Algebra: A Modern Introduction by David | Poole (which I picked up rather randomly in a library clearance | sale for $2). It tackles most subjects from both algebraic and | geometric perspectives, so from the visual aspect it might fit. | What's particularly useful about it relative to HN is it leans | into computational applications pretty heavily. | | For example, if some particular method is computationally | efficient relative to others, the text makes a note of it, and | has lots of computational examples. Most of the examples could be | set up fairly straightforwardly with something like a Python | notebook and Numpy for matrices. It also covers things like | computational errors wrt floating-point operations when doing | vector and matrix calculations, efficient algorithms for | approximating eigenvalues of a matrix, etc. | | And!, the full text is available on archive.org with a free | account: | | https://archive.org/details/linearalgebramod0000pool | rileytg wrote: | I would highly recommend starting with khan academy. it's pretty | visual and worked great for me- a largely visual learner. | aurnik wrote: | Fun series on learning practical linear algebra from a robotics | engineer: https://youtu.be/FKs1XhlrZDw | | I don't remember how I found this guy but watching him feels more | like learning from a friend who's extremely knowledgeable about | linear algebra rather than sitting in a university course. | mch82 wrote: | "Linear Algebra" on Wikibooks may be worth a look (and consider | helping to make it better if it's not useful enough yet) | https://en.m.wikibooks.org/wiki/Linear_Algebra | ibobev wrote: | http://immersivemath.com/ila/index.html | i-das wrote: | Introduction to Applied Linear Algebra - Vectors, Matrices, and | Least Squares : https://web.stanford.edu/~boyd/vmls/ | rglullis wrote: | Off-topic, I know... but let's not propagate the idea that there | is such a thing as "visual learners": | https://www.veritasium.com/videos/2021/7/9/the-biggest-myth-... | fugalfervor wrote: | There's no such thing as a visual learner | bryanrasmussen wrote: | I guess I'm going to have to call up that psychologist that | gave my daughter that evaluation and give him a piece of your | mind! But aside from the flat statement do you have anything to | back it up? | thaumasiotes wrote: | > But aside from the flat statement do you have anything to | back it up? | | Well, here's a comment from elsewhere in the thread: | | > If you can do one thing now, watch this Veritasium video to | disprove the myth that you're a visual learner: | https://youtu.be/rhgwIhB58PA. | | ( https://news.ycombinator.com/item?id=31707314 ) | | I haven't watched the video, but, like your parent comment, I | was already aware that "learning styles" was a research area | supported almost exclusively by fraud. If you want more | links, you can find them pretty easily through | https://en.wikipedia.org/wiki/Learning_styles#Criticism . | Diris wrote: | Veritasium has a very good video on the subject.[0] Sources | are in the description but I might as well post them here. | | Pashler, H., McDaniel, M., Rohrer, D., & Bjork, R. (2008). | Learning styles: Concepts and evidence. Psychological science | in the public interest, 9(3), 105-119. -- | https://ve42.co/Pashler2008 | | Willingham, D. T., Hughes, E. M., & Dobolyi, D. G. (2015). | The scientific status of learning styles theories. Teaching | of Psychology, 42(3), 266-271. -- https://ve42.co/Willingham | | Massa, L. J., & Mayer, R. E. (2006). Testing the ATI | hypothesis: Should multimedia instruction accommodate | verbalizer-visualizer cognitive style?. Learning and | Individual Differences, 16(4), 321-335. -- | https://ve42.co/Massa2006 | | Riener, C., & Willingham, D. (2010). The myth of learning | styles. Change: The magazine of higher learning, 42(5), | 32-35.-- https://ve42.co/Riener2010 | | Husmann, P. R., & O'Loughlin, V. D. (2019). Another nail in | the coffin for learning styles? Disparities among | undergraduate anatomy students' study strategies, class | performance, and reported VARK learning styles. Anatomical | sciences education, 12(1), 6-19. -- | https://ve42.co/Husmann2019 | | Snider, V. E., & Roehl, R. (2007). Teachers' beliefs about | pedagogy and related issues. Psychology in the Schools, 44, | 873-886. doi:10.1002/pits.20272 -- https://ve42.co/Snider2007 | | Fleming, N., & Baume, D. (2006). Learning Styles Again: | VARKing up the right tree!. Educational developments, 7(4), | 4. -- https://ve42.co/Fleming2006 | | Rogowsky, B. A., Calhoun, B. M., & Tallal, P. (2015). | Matching learning style to instructional method: Effects on | comprehension. Journal of educational psychology, 107(1), 64. | -- https://ve42.co/Rogowskyetal | | Coffield, Frank; Moseley, David; Hall, Elaine; Ecclestone, | Kathryn (2004). -- https://ve42.co/Coffield2004 | | Furey, W. (2020). THE STUBBORN MYTH OF LEARNING STYLES. | Education Next, 20(3), 8-13. -- https://ve42.co/Furey2020 | | Dunn, R., Beaudry, J. S., & Klavas, A. (2002). Survey of | research on learning styles. California Journal of Science | Education II (2). -- https://ve42.co/Dunn2002 | | [0] The Biggest Myth In Education | https://www.youtube.com/watch?v=rhgwIhB58PA | atty wrote: | I am no expert just a curious outsider, so take this with a | large grain of salt, but it is my understanding that that's | one of the most pernicious misconceptions even in practicing | psychologists, but that the current high quality research | suggests the learning styles theory is flawed at best and | wrong at worst. This article is ~8 years old but I don't | think anything has quantitatively changed the conclusions | over the intervening years. | | https://sciencebasedmedicine.org/brain-based-learning- | myth-v... | [deleted] | [deleted] | bajsejohannes wrote: | I really liked Linear Algebra And Its Applications by David C | Lay, although it seems that more people dislike it. I believe | it's a pretty common book for college intro courses. It does | illustrate everything pretty well if I remember correctly. | | Perhaps a game development book is even more visual? I haven't | read it (yet), but this book is getting recommendations: | https://gamemath.com/book/ | jimhefferon wrote: | I have a text at https://hefferon.net/linearalgebra/index.html. | It is aimed at beginners. It comes with perhaps two dozen | exercises per lecture along with complete worked answers to every | question, with videos of the lectures, a lab manual using Sage, | and some other ancillaries. | | Like others here I recommend 3B1B, which may be what you are | looking for visually, but whatever you end up with it is | absolutely crucial that you do exercises. Do many of them. It is | the only way to get better. | haneefmubarak wrote: | Personally I liked the No Bullshit Guide to Linear Algebra. It | kind of builds up things slowly and in a conversational manner, | but you can also skip thru pretty quickly if you just need a | reference. | | I don't think I've been able to find any particularly good visual | LinAlg books - most of what you're trying to achieve is actually | quite abstract and I found the classic books a little confusing. | | As an addendum - if you live stateside, classes at community | colleges may be quite inexpensive and fairly approachable. | nsv wrote: | Second to No Bullshit Guide to Linear Algebra. It's well | written, has plenty of practice problems, and an interesting | applications section. | kqr2 wrote: | https://minireference.gumroad.com/l/noBSLA | | The author is also on HN: | | https://news.ycombinator.com/user?id=ivan_ah | thunkle wrote: | I'm going through this right now. It's really great at giving | refreshers and not assuming you know anything. | maerF0x0 wrote: | +1 | seltzered_ wrote: | Not my interest but some bookmarks : | | http://betterexplained.com/articles/linear-algebra-guide/ | | http://immersivemath.com/ila/index.html | | https://www.scribd.com/document/376657416/Linear-Algebra-in-... | whatsakandr wrote: | Highly recommend 3blue1brown's essence of linear algebra playlist | as a supplement to anything you do. I "knew" linear before | watching this playlist, now I know it. Link: | https://youtube.com/playlist?list=PL0-GT3co4r2y2YErbmuJw2L5t... | ycdavidsmith wrote: | "Introduction to Linear Algebra" by Gilbert Strang is the book. | Recommend getting a used older edition as not much has changed. | | His course at MIT is legendary, completely available online | https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010... | | And there's so much good linear algebra stuff on YouTube from | 3brown1blue. | | If you can do one thing now, watch this Veritasium video to | disprove the myth that you're a visual learner: | https://youtu.be/rhgwIhB58PA. | lamename wrote: | The point of the hook statement "You are not a visual learner" | in the Veritasium video is not to "disprove the myth that | you're a visual learner." | | The point is that there's little evidence behind different | people having different learning styles, and that in general | everyone is every "style". | | This implies that vision, in addition to many other sensory | modalities, is useful. As you point out, the utility of of 3b1b | is in line with this point. | tptacek wrote: | Just chiming in to say that you can dive directly into Strang's | Youtube lecture series, without a book or anything else; like, | an immediate next step you could take if you wanted to is just | to pull up his first lecture right now and watch it. (I mostly | watched him at 2.5x speed). | notfed wrote: | Also, Khan Academy is an excellent supplement for parts you | find confusing. | jjtheblunt wrote: | https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefet... | | that book, i think, is fantastic. i was a TA for a graduate (and | undergraduate) level course using it in Urbana-Champaign around | 25 years ago. It's just a great book. | gnicholas wrote: | Several commenters have mentioned 3B1B and other youtube videos. | I'm curious about the suggested ordering: should one watch the | videos to get an intuitive sense of things, then proceed with a | textbook/practice questions? Or would it be better to struggle | through a textbook/problems, and then watch videos to crystalize | concepts after you've primed your brain a bit? | | I realize the answers will differ for different | people/situations, but I'd be curious to know what has or hasn't | worked for others. | enhdless wrote: | I recommend _The Manga Guide to Linear Algebra_! I read it the | summer before college and their visuals and analogies really | helped me grasp basic concepts. | dragontamer wrote: | I disagree. I personally found that one to be a poorly written | "Manga Guide". (Manga Guide to SQL was a good one, but there | really weren't as many good analogies for Linear Algebra). | | A lot of the "examples" were "This is complicated and abstract, | so we'll just say it is and go to textbook form". | belfalas wrote: | I am indeed here posting my original question after first | trying the Manga Guide to Linear Algebra and finding it was | not what I was looking for. Where I wanted visual explanation | they went to textbook definitions, not helpful. A few | illustrations in the book I did think were valuable so it | wasn't a total loss. | wrycoder wrote: | LA is about vectors and rotations and stretches of vectors, | which is what happens when you multiply a vector by a | matrix. That's what you will be visualizing. | | Try the Kahn videos, then watch the 3B1B videos, which are | very visual, but somewhat advanced. Or, watch both of them | several times in parallel. | FastMonkey wrote: | 3blue1brown is fantastic. Every once in a while I'll be like | "what is the intuition behind determinants again?", and boom, | there's a thoughtful and concise video on it. | | However, there's no magic bullet that will let you learn linear | algebra in a couple of hours. At some point you have to sit down | and work to figure it out. The field has university departments | researching it, so there's a lot more to it than just multiplying | mxp by pxn matrices. | | I don't know what you mean exactly by beginner, but assuming you | have some level of mathematical maturity, UT Austin has an edx | course you can audit for free, "Linear Algebra Foundations to | Frontiers", and FastAI also have a pretty good free video | series/course on it too. | lamename wrote: | I've found the "_ for Dummies" series to be quite clear for math. | I used Linear Algebra for Dummies to brush up on some concepts | recently to solve a problem at work. | graycat wrote: | I'll get you a start here: | | For the _graphical_ part, start with, say, (3,7). Regard that as | the coordinates in the standard X-Y coordinate system of a | _point_ on the plane. So, the X coordinate is the 3 and the Y | coordinate is the 7. You could get out some graph paper and plot | the thing. So, more generally, given two numbers x and y, (x,y) | is the coordinates of a point in the plane. We call (x,y) a | _vector_ and imagine that it is an arrow from the origin to an | arrow head at point (x,y). Then we can imagine that we can slide | the vector around on the plane, keeping its length and direction | the same. | | What we did for the plane and X-Y we could do for space with | X-Y-Z. So, there a vector would have three coordinates. Ah, call | them _components_. | | Now in linear algebra, for a positive integer n, we could have a | vector with n components. For geometric intuition, what we saw in | X-Y or X-Y-Z is usually enough. | | We can let R denote the set of real numbers. Then R^n denotes the | set of all of the vectors with n components that are real | numbers. Our R^n is the leading example of a _vector space_. | Sometimes it is good to permit the components to be complex | numbers, but that is a little advanced. And the components could | be elements of some goofy finite _field_ from abstract algebra, | but that also is a bit advanced. | | Here we just stay with the real numbers, elements of R. | | Okay, suppose someone tells us | | ax + by = s | | cx + dy = t | | where the a, b, c, d, s, t are real numbers. | | We want to know what the x and y are. Right, we can think of the | vector (x,y). Without too much work we can show that, depending | on the _coefficients_ a, ..., t, the set of all (x,y), that fits | the two equations has none, one, or infinitely many (points, | vectors) solutions. | | So, that example is two equations in two unknowns, x and y. Well, | for positive integers m and n, we could have m equations in n | unknowns. Still, there are none, one, or infinitely many | solutions. | | C. F. Gauss gave us _Gauss elimination_ that lets us know if | none, one, or infinitely many, find the one, or generate as many | as we wish of the infinite. | | We can multiply one of the equations by a number and add the | resulting equation to one of the other equations. We just did an | _elementary row operation_ , and you can convince yourself that | the set of all solutions remains the same. So, Gauss elimination | is to pick elementary row operations that make the pattern of | coefficients have a lot of zeros so that we can by inspection | read off the none, one, or infinitely many. Gauss elimination is | not difficult or tricky and programs easily in C, Fortran, .... | | Quite generally in math, if we have a function f, some numbers a | and b, and some _things_ , of high generality, e.g., our vectors, | and it is true that for any a, b and things x and y | | f(ax + by) = af(x) + bf(y) | | then we say that function f is _linear_. Now you know why our | subject is called _linear algebra_. | | A case of a linear function is Schroedinger's equation in quantum | mechanics, and linear algebra can be a good first step into some | of the math of quantum mechanics. | | Let's see why those equations were _linear_ : Let | | f(x,y) = ax + by | | Then | | f[ c(x,y) + d(u,v)] | | = f[ (cx, cy) + (du,dv) ] | | = f(cx + du, cy + dv) | | = a(cx + du) + b(cy + dv) | | = c(ax) + d(au) + c(by) + d(bv) | | = c(ax + by) + d(au + bv) | | = cf(x,y) + df(u,v) | | Done! | | This _linearity_ is mostly what makes linear algebra get its | mathematical theorems and its utility in applications. | | We commonly regard the plane with coordinates X-Y as _2 | dimensional_ and space with coordinates X-Y-Z as _3 dimensional_. | If we study _dimension_ carefully, then the 2 and 3 are correct. | Similarly R^n is n dimensional. | | We can write | | ax + by = s | | cx + dy = t | | as x(a,c) + y(b,d) = (s,t) | | So, (a,c), (b,d), and (s,t) are vectors, and x and y are | coefficients that let us write vector (s,t) as a linear | combination of the two vectors (a,c) and (b,d). | | Apparently the _superposition_ in quantum mechanics is closely | related to this linear combination. | | Well suppose these two vectors (a,c) and (b,d) can be used in | such a linear combination to get any vector (s,t). Then, omitting | some details, (a,c) and (b,d) _span_ all of R^2, are _linearly | independent_ , and form a _basis_ for the vector space R^2. | | Sure, the usual basis for R^2 is just | | (1,0) | | (0,1) | | And that our basis has two vectors is because R^2 is 2 | dimensional. Works the same in R^n -- n dimensional and a basis | has n vectors that are linearly independent. | | Now for some geometric intuition, given vectors in R^3 (x,y,z) | and (u,v,w), then for coefficients a and b, the set of all | | a(x,y,z) + b(u,v,w) | | forms, depending on the two vectors, a point, a line, or a plane | through (0,0,0) -- usually a plane and, thus, a vector _subspace_ | of dimension 0, 1, 2, usually 2. | | And this works in R^n: We can have vector subspaces of dimension | 0, 1, ..., n. For a subspace V of dimension m, 1 <= m <= n, there | will be a basis of m linearly independent vectors in subspace V. | | Let's explain matrix notation: Back to | | ax + by = s | | cx + dy = t | | On the left side, let's rip out the x and y and write the rest as | /a b\ | | \c d/ | | So, this _matrix_ has two rows and two columns. Let 's call this | matrix A. For positive integers m and n, we can have a matrix | with m rows and n columns and call it an m x n (pronounced m by | n) matrix. | | The (x,y) we can now call a 1 x 2 matrix. But we really want its | _transpose_ , 2 x 1 as /x\ | | | \y/ | | Let's call this matrix v. | | We want to define the matrix product Av | | We define it to be just what we saw in | | ax + by = s | | cx + dy = t | | That is, Av is the transpose of (s,t). | | If we have a vector u and coefficients a and b and define matrix | addition in the obvious way, we can have A(au | + bv) = aAu + bAv = a(Au) + b(Av) = | (aA)u + (bA)v | | that is, we have some (associativity), so that A acts like a | linear function. Right, the subject is linear algebra. | | And matrix multiplication is associative, and the usual proof is | just an application of the interchange of summation signs for | finitely many terms. | | We can define the length of a vector and the angle between two | vectors. then multiplying two vectors by an _orthogonal_ matrix U | does not change the length or angle of two vectors. | | Then for any orthogonal matrix U, all it does is reflect and/or | make a rigid rotation. | | We can also have a _symmetric, positive definite_ matrix S. What | S does is stretch a sphere into an ellipsoid (the 3 dimensional | case does provide good intuition). Then A can be written as SU. | That is, all A can do is rotate and reflect and then move a | sphere into an ellipsoid. That is the _polar decomposition_ and | is the key to much of the most advanced work in linear algebra. | Turns out, once we know more about orthogonal and symmetric | matrices, the proof is short. | | That's enough for a fast introduction! | gadrev wrote: | I believe I used this back in the day when preparing the LA | course at uni, A First Course in Linear Algebra: | http://linear.ups.edu/download.html . | | Also I'm sure it's been mentioned here already, but the MIT | Linear Algebra course by Gilbert Strang was absolutely | phenomenal. Really made it click for me. | tptacek wrote: | This video about "learning styles" and the idea that there's no | such thing as a "visual learner" is all over the thread: | | https://www.youtube.com/watch?v=rhgwIhB58PA&feature=youtu.be | | It's worth calling out that the video itself doesn't support some | of the blunt arguments being made here. The point of the video is | that it's likely that everyone does better with a _multimodal_ | approach. It thus remains reasonable to seek out books that do a | good job with visual representations! No visual components, or, | worse, bad diagrams are, according to the video, an impediment to | everyone, not just people who have used a disfavored term in | their Ask HN question. :) | | I like 3Blue1Brown as much as everyone else, it's an achievement | and kind of a joy to watch, but my experience was that, after | many go-rounds over the years, the thing that made any of this | actually stick was doing exercises. I tend to bang the less | abstract ones out in Sage: https://www.sagemath.org (you have to | do a little bit of extra work to make sure Sage isn't doing too | much of the work for you. | | I'm a fan of Strang's approach. But I'm bad at linear algebra, | so, grain of salt. | scott01 wrote: | Lots of great suggestions here! The original question was | probably answered already, but nevertheless I'd like to humbly | leave a link to the article "So You Want to Study Mathematics..." | by Susan Rigetti [1] -- it covers not only linear algebra but | other parts of math as well. Quite inspirational, I'd say. | | [1]: https://www.susanrigetti.com/math | rscho wrote: | If you want something hands on: "coding the matrix" | navbaker wrote: | I'll second this book, tons of very practical exercises to help | you understand what's happening for every main concept. | gmw wrote: | Although it's not a book, a good series on YouTube is 3Blue1Brown | Essence of Linear Algebra. That explains it in a very visual way. | That, in addition to Linear Algebra and its applications by | Gilbert Strang, would be a strong mix. I would also recommend | 3000 solved problems in Linear Algebra by Seymour Lipschutz as a | strong foundation in linear algebra requires practice. | aidos wrote: | Essence of linear algebra is an absolutely wonderful series. It | gave me an intuition of the subject in a matter of hours in way | years of university didn't do. | | https://youtu.be/fNk_zzaMoSs | lagrange77 wrote: | Yes. The moment, when the background grid gets distorted by | the matrix. Really helped me to calibrate my mental models. | threatofrain wrote: | It should be noted that the sum of the 3B1B videos is like 2 | hours, and that Grant himself says that these videos are for | summarizing and providing intuition _after_ you have already | taken the course. ___________________________________________________________________ (page generated 2022-06-11 23:00 UTC)