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Ask HN: Best beginner friendly linear algebra book?

259 points by belfalas 4y ago

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!

rscho 4y ago
If you want something hands on: "coding the matrix"
- navbaker 4y ago
  I’ll second this book, tons of very practical exercises to help you understand what’s happening for every main concept.
bajsejohannes 4y ago
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/ https://gamemath.com/book/
ycdavidsmith 4y ago
“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/ 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 https://youtu.be/rhgwIhB58PA.
- notfed 4y ago
  Also, Khan Academy is an excellent supplement for parts you find confusing.
- lamename 4y ago
  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 4y ago
  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).
- trenchgun 4y ago
  Gilbert Strang also has a new book, Linear Algebra for Everyone. I am going through it now, and it is very nice.
haneefmubarak 4y ago
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 4y ago
  Second to No Bullshit Guide to Linear Algebra. It's well written, has plenty of practice problems, and an interesting applications section.
- maerF0x0 4y ago
  +1
  - cleandreams 4y ago
    I liked it also.
- kqr2 4y ago
  https://minireference.gumroad.com/l/noBSLA https://minireference.gumroad.com/l/noBSLA The author is also on HN: https://news.ycombinator.com/user?id=ivan_ah https://news.ycombinator.com/user?id=ivan_ah
- thunkle 4y ago
  I'm going through this right now. It's really great at giving refreshers and not assuming you know anything.
- ivansavz 4y ago
  @haneefmubarak Thx for the plug! For everyone who might be interested, here is an extended preview of the book here: https://minireference.com/static/excerpts/noBSLA_v2_preview.pdf https://minireference.com/static/excerpts/noBSLA_v2_preview.... and you can also download the standalone LA concept map here: https://minireference.com/static/conceptmaps/linear_algebra_concepts.pdf https://minireference.com/static/conceptmaps/linear_algebra_... If you prefer video explanations, I've got some of those up on youtube that given an overview of the topics in the book, see https://www.youtube.com/watch?v=2G3PmEZI6n8&list=PLGmu4KtWiH6-F2JlLVvCTsMR91I4ix0_z&index=1 https://www.youtube.com/watch?v=2G3PmEZI6n8&list=PLGmu4KtWiH... I highly recommend getting a print copy because it will help you concentrate on the material and read deeply without distractions. I also have a free-PDF-if-with-purchase-of-print-copy policy, so if you send get the print version and send me an email I'll hook you up with the PDF for free (PDF good for reference and for searching). Happy to take questions about anything related to the book. Post here or send email. Contacts in profile.
enhdless 4y ago
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 4y ago
  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 4y ago
    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 4y ago
      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.
      - dragontamer 4y ago
        This is why I asked "what field are you learning Linear Algebra?". Elsewhere, I've discovered that this poster is going into image processing, which is likely "signals and systems" linear algebra. In signals and systems, your vectors can have infinite dimension, and these infinite-dimension vectors Fourier-transform into other infinite dimension vectors under a new basis. Any field with more than "3 dimension" vectors / matricies is very difficult to visualize geometrically. Trying to do so is counter-productive to the understanding of the field. This geometric interpretation is really useful in graphics programming / 3d animation however. --------- Or perhaps a more concrete example... your "visualize the matrix in X dimensions" advice just doesn't cut it if you're dealing with an 8x8 matrix JPEG DCT coefficient matrix (https://en.wikipedia.org/wiki/JPEG#Discrete_cosine_transform https://en.wikipedia.org/wiki/JPEG#Discrete_cosine_transform), unless you can imagine 8-diemsional space in your brain. On the other hand, imagining the 8x8 matrix as 64 linearly-independent "Basis" to your 64-dimension discrete signal is... easier. (Well... for a definition of easier at least). And the transform from time domain into Fourier domain is a transformation in basis that contains the same information.
gmw 4y ago
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 4y ago
  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 https://youtu.be/fNk_zzaMoSs
  - lagrange77 4y ago
    Yes. The moment, when the background grid gets distorted by the matrix. Really helped me to calibrate my mental models.
- threatofrain 4y ago
  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.
  - edanm 4y ago
    I'd also add that it's probably good to watch it before as well, to give intuitions around things you'll later learn rigorously.
  - greymalik 4y ago
    By “the course” do you mean Strang’s MIT OCW class or something else?
harshreality 4y ago
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 4y ago
What are you learning for? I'm in the industry learning for work in medical image visualization.
nojito 4y ago
Nothing holds a candle to https://web.stanford.edu/~boyd/vmls/ https://web.stanford.edu/~boyd/vmls/ Applied learning is the best way to learn linear algebra.
dragontamer 4y ago
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.
- belfalas 4y ago
  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.
hn_version_0023 4y ago
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 4y ago
  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 4y ago
    Thank you kindly! Its always nice to learn you’re more prepared than you supposed!
    - nextos 4y ago
      Yeah, read Basic Mathematics by Lang. Covers all precalculus you need in a rigorous way.
- wrycoder 4y ago
  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.
  - hn_version_0023 4y ago
    That’s an excellent recommendation, thank you.
aurnik 4y ago
Fun series on learning practical linear algebra from a robotics engineer: https://youtu.be/FKs1XhlrZDw 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.
ibobev 4y ago
http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html
i-das 4y ago
Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares : https://web.stanford.edu/~boyd/vmls/ https://web.stanford.edu/~boyd/vmls/
jimhefferon 4y ago
I have a text at https://hefferon.net/linearalgebra/index.html 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.
lamename 4y ago
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.
jerDev 4y ago
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
ThatGeoGuy 4y ago
"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.
fugalfervor 4y ago
There's no such thing as a visual learner
- fugalfervor 4y ago
  I regret making this post, with no context or explanation, and I'd like to avoid making the same mistake in the future. There is always more room to grow.
- bryanrasmussen 4y ago
  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 4y ago
    > 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://youtu.be/rhgwIhB58PA. ( https://news.ycombinator.com/item?id=31707314 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 https://en.wikipedia.org/wiki/Learning_styles#Criticism .
  - Diris 4y ago
    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 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 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 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 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 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 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 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 https://ve42.co/Rogowskyetal Coffield, Frank; Moseley, David; Hall, Elaine; Ecclestone, Kathryn (2004). — https://ve42.co/Coffield2004 https://ve42.co/Coffield2004 Furey, W. (2020). THE STUBBORN MYTH OF LEARNING STYLES. Education Next, 20(3), 8-13. — https://ve42.co/Furey2020 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 https://ve42.co/Dunn2002 [0] The Biggest Myth In Education https://www.youtube.com/watch?v=rhgwIhB58PA https://www.youtube.com/watch?v=rhgwIhB58PA
whatsakandr 4y ago
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-GT3co4r2y2YErbmuJw2L5tW4Ew2O5B https://youtube.com/playlist?list=PL0-GT3co4r2y2YErbmuJw2L5t...
photochemsyn 4y ago
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 https://archive.org/details/linearalgebramod0000pool
FastMonkey 4y ago
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 m×p by p×n 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.
saeranv 4y ago
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_Linear_Algebra 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 4y ago
  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.
graycat 4y ago
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!
axegon_ 4y ago
One which was posted here is an absolute masterpiece if you ask me: https://news.ycombinator.com/item?id=24892907 https://news.ycombinator.com/item?id=24892907
rileytg 4y ago
I would highly recommend starting with khan academy. it’s pretty visual and worked great for me- a largely visual learner.
rglullis 4y ago
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-in-education https://www.veritasium.com/videos/2021/7/9/the-biggest-myth-...
- djmips 4y ago
  I didn't find veritasium's video convincing.
  - rglullis 4y ago
    Scientific, replicable studies showing no correlation between "preferred learning styles" and effectiveness of learning, vs how the whole VARK theory which was built and promoted by an educator who never even tried to test the hypothesis? If that is not convincing to you, what is?
jjtheblunt 4y ago
https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefethen/dp/0898713617 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.
tptacek 4y ago
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 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 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.
throwaway81523 4y ago
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.
isaacimagine 4y ago
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_2017-09-04.pdf https://www.math.brown.edu/streil/papers/LADW/LADW_2017-09-0... [1]: https://www.amazon.com/Linear-Algebra-Its-Applications-5th/dp/032198238X https://www.amazon.com/Linear-Algebra-Its-Applications-5th/d... — PDFs exist. [2]: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
- Syzygies 4y ago
  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 4y ago
    >I know Sheldon; he believes it's a crime to teach people determinants. Any way to explain to a lay person why?
    - mseri 4y ago
      The book is open access (https://linear.axler.net/ 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.
gadrev 4y ago
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 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.
krosaen 4y ago
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/ https://math.mit.edu/~gs/everyone/
mch82 4y ago
“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 https://en.m.wikibooks.org/wiki/Linear_Algebra
seltzered_ 4y ago
Not my interest but some bookmarks : http://betterexplained.com/articles/linear-algebra-guide/ http://betterexplained.com/articles/linear-algebra-guide/ http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html https://www.scribd.com/document/376657416/Linear-Algebra-in-4-Pages-pdf https://www.scribd.com/document/376657416/Linear-Algebra-in-...
gnicholas 4y ago
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.
- rramadass 4y ago
  Videos and Books are Orthogonal. You switch as needed to reinforce understanding. That said, i always start with a book since they tend to be comprehensive and allow me to study at my own pace. They force me to imagine and think which i don't get from any video (which is spoon-feeding at the author's pace).
scott01 4y ago
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 https://www.susanrigetti.com/math
hprotagonist 4y ago
Oldie but a goodie: Herb Gross’ stuff stands up. https://ocw.mit.edu/courses/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/ https://ocw.mit.edu/courses/res-18-008-calculus-revisited-co...
nojs 4y ago
I remember Linear Algebra and its Applications by David Lay was very approachable and had good visuals. It’s a small book that sticks to the main points.
SneakyTornado29 4y ago
Math 51 Stanford
Iwan-Zotow 4y ago
Alan Macdonald Linear and Geometric Algebra
djmips 4y ago
belfalas you probably feel you don't need another one but this is a great book that is friendly for visual learners (even though apparently they don't exist!) Practical Linear Algebra - Gerald Farin & Dianne Hansford. IIRC, the first edition was based on a work designed to educate design workers at a car firm who either didn't have the background or needed shoring up. So from the beginning it was aimed at the common person. It has since been developed over the years to it's current form. All I know is this book has great illustrations / great analogies and was written in a way that worked _for me_ where other books did not and was a springboard to more advanced texts which as you can tell by the comments there are many. I highly reccomend. http://www.farinhansford.com/books/pla/ http://www.farinhansford.com/books/pla/
- djmips 4y ago
  For my favourite math YouTuber besides 3B1B as is mentioned all over the comments, I really really like Steve Brunton at UW. I can't believe how much I've learned from his presentations on YouTube which led me to his book 'Data-Driven Science And Engineering' with J. Nathan Kurtz. Excellent book and excellent YouTube videos. http://www.databookuw.com/ http://www.databookuw.com/ https://www.youtube.com/channel/UCm5mt-A4w61lknZ9lCsZtBw https://www.youtube.com/channel/UCm5mt-A4w61lknZ9lCsZtBw
- a_e_k 4y ago
  As a visual thinker and graphics practioner, I'll second this one. This is the linear algebra text that I keep on the bookcase near my desk.
- rramadass 4y ago
  Seconded. The full name of the book is; Practical Linear Algebra: A Geometry Toolbox. The preface states The subtitle of this book is A Geometry Toolbox; this is meant to emphasize that we approach linear algebra in a geometric and algorithmic way (The first edition was named "The Geometry Toolbox for Graphics and Modeling"). Pair it with Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares by Boyd and Vandenberghe (https://web.stanford.edu/~boyd/vmls/ https://web.stanford.edu/~boyd/vmls/) and you are golden.
iamcreasy 4y ago
Did you take any college course on this? In class, I've found Linear Algebra to be confusing as there are many ways to say the same things and ideas are connected to each other like a giant web. I would not be able to follow along if I did not have my instructor who is willing to pause and explain it to me. I recommend enrolling in a class, explain the instructor about your goal and s/he can find topics and walk you through them.
boloust 4y ago
There's many good recommendations for books here. But if you're a developer without much mathematics experience, I think you can harness your dev experience to learn far more efficiently than a standard approach. Basically, go through the exercises in the book, but use numpy, Julia, MATLAB, or similar. Learn the "Matrix API" back to front. I've successfully taught a number of devs without a strong mathematics background with this method. The fast feedback loop this approach affords allows a good intuition to be built up very quickly.
johnamata 4y ago
I used a combination of books, and all of them were mentioned in this thread. But the best resource that i've used as a self-learner werent books but udemy courses from a math professor Hania Uscka-Wehlou: https://www.udemy.com/user/hania-uscka-wehlou/ https://www.udemy.com/user/hania-uscka-wehlou/ Great for visual learners: full of colours, highlights relevant parts of the equations, visualisations, etc.
FabHK 4y ago
Not sure whether it's been said, but: there are not shortcuts. Sure, there are somewhat more and somewhat less intuitive texts, but there are no shortcuts. So I somewhat dislike the question.
bmitc 4y ago
There's a lot of great responses, but I really like Linear Algebra by Harold Edwards. He's an excellent writer and takes an algorithmic approach in his approach. It's pretty short and designed for one-semester, and you could learn a lot from coding the various algorithms.
trenchgun 4y ago
Gilbert Strang: Linear Algebra for Everyone
- tomjakubowski 4y ago
  Read it with the OCW lectures! "Combinations of columns"
aero-glide2 4y ago
Coding the Matrix
hgomersall 4y ago
Linear and Geometric Algebra, by Alan MacDonald. http://www.faculty.luther.edu/~macdonal/laga/ http://www.faculty.luther.edu/~macdonal/laga/ It's a great and very geometric presentation of linear algebra that takes it through to geometric algebra. A very nice presentation of a coherent set of geometric techniques.
gtsnexp 4y ago
"Linear Algebra done right" by Sheldon Axler. He also has a beautiful series for videos on YouTube that can serve as a companion to the book.
penguin_booze 4y ago
My recommendation is always Pavel Grinfeld's youtube 3-part playlists, starting from https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS1....
zhamisen 4y ago
I think you'll enjoy Matrix Analysis and Applied Linear Algebra from Carl Meyer: http://matrixanalysis.com/ http://matrixanalysis.com/ Almost every topic is illustrated with an application or illustrative example, and the applications are very interesting.
agumonkey 4y ago
I bought "Linear Algebra with Applications by Gareth Williams" 5th or 6th edition on ebay for 6 bucks. It's very simple and operative (manual matrix calc so you don't stay stuck in fuzzy terminology for too long). It's so easy I'm not even sure it's proper undergrad level. But it deeply unlocked my brain on the topic..

259 points by belfalas 4y ago

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!

rscho 4y ago

If you want something hands on: "coding the matrix"

navbaker 4y ago
I’ll second this book, tons of very practical exercises to help you understand what’s happening for every main concept.

bajsejohannes 4y ago

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/ https://gamemath.com/book/

ycdavidsmith 4y ago

“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/ 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 https://youtu.be/rhgwIhB58PA.

notfed 4y ago
Also, Khan Academy is an excellent supplement for parts you find confusing.
lamename 4y ago
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 4y ago
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).
trenchgun 4y ago
Gilbert Strang also has a new book, Linear Algebra for Everyone. I am going through it now, and it is very nice.

haneefmubarak 4y ago

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 4y ago
Second to No Bullshit Guide to Linear Algebra. It's well written, has plenty of practice problems, and an interesting applications section.
maerF0x0 4y ago
+1
- cleandreams 4y ago
  I liked it also.
kqr2 4y ago
https://minireference.gumroad.com/l/noBSLA https://minireference.gumroad.com/l/noBSLA The author is also on HN: https://news.ycombinator.com/user?id=ivan_ah https://news.ycombinator.com/user?id=ivan_ah
thunkle 4y ago
I'm going through this right now. It's really great at giving refreshers and not assuming you know anything.
ivansavz 4y ago
@haneefmubarak Thx for the plug! For everyone who might be interested, here is an extended preview of the book here: https://minireference.com/static/excerpts/noBSLA_v2_preview.pdf https://minireference.com/static/excerpts/noBSLA_v2_preview.... and you can also download the standalone LA concept map here: https://minireference.com/static/conceptmaps/linear_algebra_concepts.pdf https://minireference.com/static/conceptmaps/linear_algebra_... If you prefer video explanations, I've got some of those up on youtube that given an overview of the topics in the book, see https://www.youtube.com/watch?v=2G3PmEZI6n8&list=PLGmu4KtWiH6-F2JlLVvCTsMR91I4ix0_z&index=1 https://www.youtube.com/watch?v=2G3PmEZI6n8&list=PLGmu4KtWiH... I highly recommend getting a print copy because it will help you concentrate on the material and read deeply without distractions. I also have a free-PDF-if-with-purchase-of-print-copy policy, so if you send get the print version and send me an email I'll hook you up with the PDF for free (PDF good for reference and for searching). Happy to take questions about anything related to the book. Post here or send email. Contacts in profile.

enhdless 4y ago

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 4y ago
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 4y ago
  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 4y ago
    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.
    - dragontamer 4y ago
      This is why I asked "what field are you learning Linear Algebra?". Elsewhere, I've discovered that this poster is going into image processing, which is likely "signals and systems" linear algebra. In signals and systems, your vectors can have infinite dimension, and these infinite-dimension vectors Fourier-transform into other infinite dimension vectors under a new basis. Any field with more than "3 dimension" vectors / matricies is very difficult to visualize geometrically. Trying to do so is counter-productive to the understanding of the field. This geometric interpretation is really useful in graphics programming / 3d animation however. --------- Or perhaps a more concrete example... your "visualize the matrix in X dimensions" advice just doesn't cut it if you're dealing with an 8x8 matrix JPEG DCT coefficient matrix (https://en.wikipedia.org/wiki/JPEG#Discrete_cosine_transform https://en.wikipedia.org/wiki/JPEG#Discrete_cosine_transform), unless you can imagine 8-diemsional space in your brain. On the other hand, imagining the 8x8 matrix as 64 linearly-independent "Basis" to your 64-dimension discrete signal is... easier. (Well... for a definition of easier at least). And the transform from time domain into Fourier domain is a transformation in basis that contains the same information.

gmw 4y ago

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 4y ago
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 https://youtu.be/fNk_zzaMoSs
- lagrange77 4y ago
  Yes. The moment, when the background grid gets distorted by the matrix. Really helped me to calibrate my mental models.
threatofrain 4y ago
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.
- edanm 4y ago
  I'd also add that it's probably good to watch it before as well, to give intuitions around things you'll later learn rigorously.
- greymalik 4y ago
  By “the course” do you mean Strang’s MIT OCW class or something else?

harshreality 4y ago

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 4y ago

What are you learning for? I'm in the industry learning for work in medical image visualization.

nojito 4y ago

Nothing holds a candle to https://web.stanford.edu/~boyd/vmls/ https://web.stanford.edu/~boyd/vmls/ Applied learning is the best way to learn linear algebra.

dragontamer 4y ago

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.

belfalas 4y ago
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.

hn_version_0023 4y ago

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 4y ago
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 4y ago
  Thank you kindly! Its always nice to learn you’re more prepared than you supposed!
  - nextos 4y ago
    Yeah, read Basic Mathematics by Lang. Covers all precalculus you need in a rigorous way.
wrycoder 4y ago
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.
- hn_version_0023 4y ago
  That’s an excellent recommendation, thank you.

aurnik 4y ago

Fun series on learning practical linear algebra from a robotics engineer: https://youtu.be/FKs1XhlrZDw 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.

ibobev 4y ago

http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html

i-das 4y ago

Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares : https://web.stanford.edu/~boyd/vmls/ https://web.stanford.edu/~boyd/vmls/

jimhefferon 4y ago

I have a text at https://hefferon.net/linearalgebra/index.html 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.

lamename 4y ago

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.

jerDev 4y ago

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

ThatGeoGuy 4y ago

"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.

fugalfervor 4y ago

There's no such thing as a visual learner

fugalfervor 4y ago
I regret making this post, with no context or explanation, and I'd like to avoid making the same mistake in the future. There is always more room to grow.
bryanrasmussen 4y ago
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 4y ago
  > 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://youtu.be/rhgwIhB58PA. ( https://news.ycombinator.com/item?id=31707314 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 https://en.wikipedia.org/wiki/Learning_styles#Criticism .
- Diris 4y ago
  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 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 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 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 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 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 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 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 https://ve42.co/Rogowskyetal Coffield, Frank; Moseley, David; Hall, Elaine; Ecclestone, Kathryn (2004). — https://ve42.co/Coffield2004 https://ve42.co/Coffield2004 Furey, W. (2020). THE STUBBORN MYTH OF LEARNING STYLES. Education Next, 20(3), 8-13. — https://ve42.co/Furey2020 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 https://ve42.co/Dunn2002 [0] The Biggest Myth In Education https://www.youtube.com/watch?v=rhgwIhB58PA https://www.youtube.com/watch?v=rhgwIhB58PA

whatsakandr 4y ago

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-GT3co4r2y2YErbmuJw2L5tW4Ew2O5B https://youtube.com/playlist?list=PL0-GT3co4r2y2YErbmuJw2L5t...

photochemsyn 4y ago

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 https://archive.org/details/linearalgebramod0000pool

FastMonkey 4y ago

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 m×p by p×n 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.

saeranv 4y ago

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_Linear_Algebra 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 4y ago
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.

graycat 4y ago

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!

axegon_ 4y ago

One which was posted here is an absolute masterpiece if you ask me: https://news.ycombinator.com/item?id=24892907 https://news.ycombinator.com/item?id=24892907

rileytg 4y ago

I would highly recommend starting with khan academy. it’s pretty visual and worked great for me- a largely visual learner.

rglullis 4y ago

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-in-education https://www.veritasium.com/videos/2021/7/9/the-biggest-myth-...

djmips 4y ago
I didn't find veritasium's video convincing.
- rglullis 4y ago
  Scientific, replicable studies showing no correlation between "preferred learning styles" and effectiveness of learning, vs how the whole VARK theory which was built and promoted by an educator who never even tried to test the hypothesis? If that is not convincing to you, what is?

jjtheblunt 4y ago

https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefethen/dp/0898713617 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.

tptacek 4y ago

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 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 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.

throwaway81523 4y ago

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.

isaacimagine 4y ago

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_2017-09-04.pdf https://www.math.brown.edu/streil/papers/LADW/LADW_2017-09-0... [1]: https://www.amazon.com/Linear-Algebra-Its-Applications-5th/dp/032198238X https://www.amazon.com/Linear-Algebra-Its-Applications-5th/d... — PDFs exist. [2]: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

Syzygies 4y ago
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 4y ago
  >I know Sheldon; he believes it's a crime to teach people determinants. Any way to explain to a lay person why?
  - mseri 4y ago
    The book is open access (https://linear.axler.net/ 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.

gadrev 4y ago

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 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.

krosaen 4y ago

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/ https://math.mit.edu/~gs/everyone/

mch82 4y ago

“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 https://en.m.wikibooks.org/wiki/Linear_Algebra

seltzered_ 4y ago

Not my interest but some bookmarks : http://betterexplained.com/articles/linear-algebra-guide/ http://betterexplained.com/articles/linear-algebra-guide/ http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html https://www.scribd.com/document/376657416/Linear-Algebra-in-4-Pages-pdf https://www.scribd.com/document/376657416/Linear-Algebra-in-...

gnicholas 4y ago

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.

rramadass 4y ago
Videos and Books are Orthogonal. You switch as needed to reinforce understanding. That said, i always start with a book since they tend to be comprehensive and allow me to study at my own pace. They force me to imagine and think which i don't get from any video (which is spoon-feeding at the author's pace).

scott01 4y ago

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 https://www.susanrigetti.com/math

hprotagonist 4y ago

Oldie but a goodie: Herb Gross’ stuff stands up. https://ocw.mit.edu/courses/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/ https://ocw.mit.edu/courses/res-18-008-calculus-revisited-co...

nojs 4y ago

I remember Linear Algebra and its Applications by David Lay was very approachable and had good visuals. It’s a small book that sticks to the main points.

SneakyTornado29 4y ago

Math 51 Stanford

Iwan-Zotow 4y ago

Alan Macdonald Linear and Geometric Algebra

djmips 4y ago

belfalas you probably feel you don't need another one but this is a great book that is friendly for visual learners (even though apparently they don't exist!) Practical Linear Algebra - Gerald Farin & Dianne Hansford. IIRC, the first edition was based on a work designed to educate design workers at a car firm who either didn't have the background or needed shoring up. So from the beginning it was aimed at the common person. It has since been developed over the years to it's current form. All I know is this book has great illustrations / great analogies and was written in a way that worked _for me_ where other books did not and was a springboard to more advanced texts which as you can tell by the comments there are many. I highly reccomend. http://www.farinhansford.com/books/pla/ http://www.farinhansford.com/books/pla/

djmips 4y ago
For my favourite math YouTuber besides 3B1B as is mentioned all over the comments, I really really like Steve Brunton at UW. I can't believe how much I've learned from his presentations on YouTube which led me to his book 'Data-Driven Science And Engineering' with J. Nathan Kurtz. Excellent book and excellent YouTube videos. http://www.databookuw.com/ http://www.databookuw.com/ https://www.youtube.com/channel/UCm5mt-A4w61lknZ9lCsZtBw https://www.youtube.com/channel/UCm5mt-A4w61lknZ9lCsZtBw
a_e_k 4y ago
As a visual thinker and graphics practioner, I'll second this one. This is the linear algebra text that I keep on the bookcase near my desk.
rramadass 4y ago
Seconded. The full name of the book is; Practical Linear Algebra: A Geometry Toolbox. The preface states The subtitle of this book is A Geometry Toolbox; this is meant to emphasize that we approach linear algebra in a geometric and algorithmic way (The first edition was named "The Geometry Toolbox for Graphics and Modeling"). Pair it with Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares by Boyd and Vandenberghe (https://web.stanford.edu/~boyd/vmls/ https://web.stanford.edu/~boyd/vmls/) and you are golden.

iamcreasy 4y ago

Did you take any college course on this? In class, I've found Linear Algebra to be confusing as there are many ways to say the same things and ideas are connected to each other like a giant web. I would not be able to follow along if I did not have my instructor who is willing to pause and explain it to me. I recommend enrolling in a class, explain the instructor about your goal and s/he can find topics and walk you through them.

boloust 4y ago

There's many good recommendations for books here. But if you're a developer without much mathematics experience, I think you can harness your dev experience to learn far more efficiently than a standard approach. Basically, go through the exercises in the book, but use numpy, Julia, MATLAB, or similar. Learn the "Matrix API" back to front. I've successfully taught a number of devs without a strong mathematics background with this method. The fast feedback loop this approach affords allows a good intuition to be built up very quickly.

johnamata 4y ago

I used a combination of books, and all of them were mentioned in this thread. But the best resource that i've used as a self-learner werent books but udemy courses from a math professor Hania Uscka-Wehlou: https://www.udemy.com/user/hania-uscka-wehlou/ https://www.udemy.com/user/hania-uscka-wehlou/ Great for visual learners: full of colours, highlights relevant parts of the equations, visualisations, etc.

FabHK 4y ago

Not sure whether it's been said, but: there are not shortcuts. Sure, there are somewhat more and somewhat less intuitive texts, but there are no shortcuts. So I somewhat dislike the question.

bmitc 4y ago

There's a lot of great responses, but I really like Linear Algebra by Harold Edwards. He's an excellent writer and takes an algorithmic approach in his approach. It's pretty short and designed for one-semester, and you could learn a lot from coding the various algorithms.

trenchgun 4y ago

Gilbert Strang: Linear Algebra for Everyone

tomjakubowski 4y ago
Read it with the OCW lectures! "Combinations of columns"

aero-glide2 4y ago

Coding the Matrix

hgomersall 4y ago

Linear and Geometric Algebra, by Alan MacDonald. http://www.faculty.luther.edu/~macdonal/laga/ http://www.faculty.luther.edu/~macdonal/laga/ It's a great and very geometric presentation of linear algebra that takes it through to geometric algebra. A very nice presentation of a coherent set of geometric techniques.

gtsnexp 4y ago

"Linear Algebra done right" by Sheldon Axler. He also has a beautiful series for videos on YouTube that can serve as a companion to the book.

penguin_booze 4y ago

My recommendation is always Pavel Grinfeld's youtube 3-part playlists, starting from https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS1....

zhamisen 4y ago

I think you'll enjoy Matrix Analysis and Applied Linear Algebra from Carl Meyer: http://matrixanalysis.com/ http://matrixanalysis.com/ Almost every topic is illustrated with an application or illustrative example, and the applications are very interesting.

agumonkey 4y ago

I bought "Linear Algebra with Applications by Gareth Williams" 5th or 6th edition on ebay for 6 bucks. It's very simple and operative (manual matrix calc so you don't stay stuck in fuzzy terminology for too long). It's so easy I'm not even sure it's proper undergrad level. But it deeply unlocked my brain on the topic..