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The Little Book of Linear Algebra
- vismit2000 1y agoRelated: Immersive Linear Algebra - https://immersivemath.com/ila/ https://immersivemath.com/ila/
- ivansavz 1y agoWow very nice. Lots of content in here, with no lengthy explanations but useful point-form intuition. The .epub has very clean math done in HTML (no images), which is a cool way to do things. I've never seen this before. I wonder what the author used to produce the .epub from the .tex?
- ivansavz 1y agoUpdate: using Sigil to look inside the .epub, I saw it was produced by `pandoc` and the math is rendered as MathML.
- michelpp 1y agoBeautiful, a great intro to and reference to core concepts. Definitely keeping this one around for mental refresh!
- snarfy 1y agoYep, same here. It is a good refresher.
- ddavis 1y agoThe organization and formatting of the single .tex file is such that one could almost read the source alone. Really nice. Also, I had no idea that GitHub did such a good job rendering the LaTeX math in markdown, it's imperfect but definitely good.
- csunoser 1y agoAt around 7.4 orthonormal basis and there after, the tex rendering stops working on the github readme preview page. Instead, it is replaced with a red error box saying: [ Unable to render expression. ] I wonder if there is an artificial limit for the amount of latex expression that can rendered per page.
- asplake 1y agoI switched to the epub at that point. Still, credit I think to github that the page renders as well as it does.
- rw_panic0_0 1y agohow is it beginner friendly, first paragraph and already an obscure formula for non math people
- cybrox 1y agoYou will never find a level of "beginner friendly" that suits everyone. I agree that this is not an ideal start - at least without any further clarification - for beginners but I think it works well for people that already known mathematical notation but not many specifics of linear algebra. Also, I don't want to be the preacher bringing this into every argument but this is one of the genuinely good uses for AI that I have found. Bringing the beginning of a beginner friendly work down to my level. I can have it explain this if I'm unsure about the specific syntax and it will convey the relevant idea (which is explained in a bit of unnecessary complexity / generality, yes) in simple terms.
- relaxing 1y agoI agree. I wouldn’t consider someone who has taken (and remembers) a course in set theory a beginner without some added qualifier. One of my pet peeves is using mathematical symbols beyond basic arithmetic without introducing them once by name. Trying to figure out what a symbol is and what branch of math it comes from is extremely frustrating.
- schoen 1y agoThere are a handful of textbooks that have a nice appendix that defines each symbol (or maybe sometimes tells you where you can go to learn more about the topic represented by the symbol!). That way they can presume that most readers are already familiar but still be helpful to those who aren't.
- CamperBob2 1y agoI haven't taken any courses in set theory, but it makes perfect sense to me. Once someone tells you that the funny script 'R' means "Real numbers", the funny E means "is an element of", and the vertical | means "given that," that's pretty much all you need to know to dive in. If those concepts cause difficulty, it probably makes sense to go back down the learning curve a bit before tackling linear algebra. Alternatively, just cut and paste the expression into any LLM and it'll explain what's what.
- barrenko 1y agoTried to pick a book to get into linear algebra recently, the experience was fairly hellish. First course this, second course that, done right, done wrong... I'd to the LADR4e route, but I don't have the proof-it chops yet...
- cybrox 1y agoI found the book "Linear Algebra" and accompanying lecture recordings by Jim Hefferon very approachable and solid. It's free, including exercises and an (also free) solutions book.
- GabriDaFirenze 1y agoI've found the "No bullshit Guide to Linear Algebra" pretty good. Could be worth checking it out. It's the one resource that has things click more for me.
- griffzhowl 1y agoI like Serge Lang's books for clarity of explanations. He has an Introduction to Linear Algebra which concisely covers the basics (265 pages in the main text), and grounds the matrix computations in the geometric interpretation. Be aware that Lang has another book, called just "Linear Algebra", which is more theoretical.
- barrenko 1y agoThis book somehow managed the escape the usual recommendations, thank you.
- rramadass 1y agoYou might want to checkout the book Practical Linear Algebra: A Geometry Toolbox by Dianne Hansford and Gerald Farin (its 1st edition was simply named The Geometry Toolbox: For Graphics and Modeling) to get an intuitive and visual introduction to Linear Algebra. Pair it with Edgar Goodaire's Linear Algebra: Pure & Applied and you can transition nicely from intuitive geometric to pure mathematical approach. The author's writing style is quite accessible. Add in Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares by Stephen Boyd et al. and you are golden. Free book available at https://web.stanford.edu/~boyd/vmls/ https://web.stanford.edu/~boyd/vmls/
- photon_lines 1y agoIf anyone is interested in a more visual or intuitive over-view, I made a mini-book on it as well a few years ago which you can find here: https://github.com/photonlines/Intuitive-Overview-of-Linear-Algebra-Fundamentals https://github.com/photonlines/Intuitive-Overview-of-Linear-...
- WillAdams 1y agoNice pairing the text with 3Blue1Brown's lectures on linear algebra! https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
- phforms 1y agoAs an autodidact who never learned this stuff at school/uni, his lectures are what made linear algebra really click for me. I can only recommend them to anyone who wants to get a visual intuition on the fundamentals of LA. What also helped me as a visual learner was to program/setup tiny experiments in Processing[1] and GeoGebra Classic[2]. - [1] https://processing.org https://processing.org - [2] https://www.geogebra.org/classic https://www.geogebra.org/classic
- lll-o-lll 1y agoNitpick: No one is a visual learner, or more correctly everyone is. Multimodal is the way, so good teachers will express the same concepts in several ‘modes’ and that helps develop the intuition.
- jonahx 1y agoThis looks very nice. Thanks for posting.
- m3047 1y agoI'd encourage the author to put a pointer to the repo in the actual doc. Maybe I should send a PR...
- andrewla 1y agoIt's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matrices and Gaussian elimination, all the while bored out of your skull, until you have the tools to really start to approach the interesting areas. Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away until one day when you're looking at a chain of linear transformations you realize that everything clicks. This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains. It's funny because it's just a pedagogical nightmare to get students to care about any of this until one day two years later it all just makes sense.
- Chinjut 1y agoWhy do you say it's practically impossible to motivate matrix multiplication? The motivation is that this represents composition of linear functions, exactly as you follow up by mentioning. It's a disservice to anyone to tell them "Well, that's the way it is" instead of telling them from the start "Look, these represent linear functions. And look, this is how they compose".
- andrewla 1y agoSure, that's a way to approach it. All you have to do is stay interested in "linear functions" long enough to get there. It's totally possible -- I got there, and so did many many many other people (arguably everyone who has applied mathematics to almost any problem has). But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fact, in one dimension it's just multiplication -- that's the only linear function and the class of scalar linear functions is completely specified by the factor that you multiple by". I stuck to it because that was what the course taught, and they wouldn't teach me multidimensional calculus without making me learn this stuff first, but it was months and years later when I suddenly found that linear functions were everywhere and I somehow magically had the tools and the knowledge to do stuff with them.
- defanor 1y agoAlways nice to see CC-licensed textbooks. This one looks fairly minimal, not including much of explanation, illustrations, or proofs; I think those are generally useful for the initial study, but this should still work as a cheat sheet, at least.
- rossant 1y agoA linear algebra course without graphics? When I learnt it at school almost 25 years ago, the teacher made schematics all the time to explain the visual intuition behind each concept. I was totally confused when he introduced the abstract definition of a vector space with the addition and scalar multiplication. Then he drew some arrows. Then it all suddenly made sense.
- eliaskickbush 1y agoHighly recommend to anyone struggling with linear algebra to check out Linear Algebra Done Right, by Sheldon Axler. Do always keep in mind that some concepts are very verbose, but truly out of necessity. If you're talking about an N by N matrix, you're naturally going to have to distinguish N^2 different elements. You can go very far without touching matrices, and actually find motivation on this abstract base before learning how it interops with matrices.
- akww 1y agoA little surprising to me this doesn’t come up the most! Excellent text, and look, you can get the 4th edition (2024) for free as a pdf at http://axler.net http://axler.net.
- russellbeattie 1y agoSomeone should convert all the examples into C code so it's more intelligible to programmers who are, let's admit, the main audience for something like this. To the best of my knowledge: Scalars are variables. Vectors are arrays. Matrices are multi dimensional arrays. Addition and multiplication is iteration with operators. Combinations are concatenation. The rest like dot products or norms are just specialized functions. But it'd be nice to see it all coded up. It wouldn't be as concise, but it'd be readable.
- CamperBob2 1y agoThat's basically what you'll get if you pick up a book on 3D game programming. However, progress will come to a halt when you get to things like determinants and eigenvalues that don't show up in core 3D graphics pipelines. You'll have to find other ways to motivate a C version of that part of the curriculum... but I agree, that's a well-worthwhile thing to ask for.
- dkga 1y agoBy the way, 3Blue1Brown’s videos on linear algebra are nothing my short of amazing. And I use linear algebra every day (I’m an economist).
- anthk 1y agoKlong/k looks ideal for this to have some practice with linear alg concepts: http://t3x.org/klong/index.html http://t3x.org/klong/index.html
- rramadass 1y agoThis was one of the reasons for me wanting to study the J programming language. I have a notion AI/ML programming might be better done using array languages. Some books for studying Mathematics using J are listed here - https://code.jsoftware.com/wiki/Books https://code.jsoftware.com/wiki/Books
- jean_lannes 1y agoAs someone who took a standard undergrad linear algebra course but never really used it in my work, what are some good ways to get acquainted with practical applications of linear algebra?
- defrost 1y agoThere were some hints upstream: https://news.ycombinator.com/item?id=45107638 https://news.ycombinator.com/item?id=45107638 Machine learning, LLMs, RSA, etc. It's generally useful for multivariate statistics, 3D flies (insects), in 3D space, clustering about a narrow slanting plane of light from a window slit are points that can be projected onto "the plane of best fit" - nominally the slanting plane of light. That right there is a geometric picture of fitting a line, a plane, a lower order manifold, to a higher order data set, the errors (distance from plane), etc. and something of what Singular Value Decomposition is about (used for image enhancement, sharpening fuzzy data, etc). The real test of applications is what kind of work do you see yourself doing? - A quick back read suggests your currently a CS student, so all unfocused potential for now (perhaps).
- JBits 1y agoA good use of linear algebra that I'm working with at the moment is the use of splines as a basis for real (vector) functions. After obtaining the matrix/vector representations you can solve for the spline coefficients (and then plot them). Linear transforms (such as rotations and displacements) in GPU graphics. Fourier series in signal processing. JPEG compression. Obtaining the best fit element in a vector space of curves given data or other constraints. Understanding autodiff in JAX. The mathematical definition of a tensor helps develop intuition for manipulating arrays/tensors in array libraries. Transition matrices of a Markov chain. PageRank.
- tekknolagi 1y agoIn case the author didn't notice, the organization name appears to be missing an L - "litte"
- zkmon 1y agoIn terms of understanding why something is like that, Linear Algebra belongs to Geometry more than Algebra. Every formula in Linear Algebra which ultimately is justified by "it's just that way" can be better justified by geometry. The algebraic formulas are like animals who lost their natural habitat and were put in a zoo, making people think that these animals evolved in the zoo itself. To give an example: A simple multiplication of two numbers is better seen as rotating one of the numbers to be perpendicular to the other and then quantifying the area/volume spanned by them. This gives vector dot product. While geometry might better address "why", algebra gets into the work of "how to do it". Mathematics in old times, like other branches of science, did not encourage "why". Instead, most stuff would say "This is how to do it, Now just do it". Algebra probably evolved to answer "how to do it" - the need to equip the field workers with techniques of calculating numbers, instead of answering their "why" questions. In this sense, Geometry is more fundamental providing the roots of concepts and connecting all equations to the real world of spatial dimensions. Physics adds time to this, addressing the change, involving human memory of the past, perceiving the change.
- hinkley 1y agoI remember how betrayed I felt when I got to calculus and realized all of those equations I learned in high school for physics were just introductory calculus. Like why did we have to learn all that the hard way?
- rramadass 1y agoNot just Linear Algebra but every branch of Science/Mathematics should be taught as much as possible using Geometry (and other visualizations) before being mapped to abstract algebraic symbols. Basic Geometry (along with simple Arithmetic) is the oldest subfield of Mathematics for the reason that most of its concepts are intuitive and feels "natural" and mappable to the "Real World" by us Humans. While abstraction via symbol manipulation is necessary to generalize and extend mathematics it should come at a later stage after we have developed some intuition of the concepts being represented by the symbols, however limited/restricted they might be. All abstraction requires some mathematical maturity which can only happen over time.
- mugamuga 1y agoIs there a version or a similar book that deals with Calculus?
- griffzhowl 1y agoSerge Lang's "Short Calculus" is clear and concise, efficiently covering the basics in ~170 pages instead of hundreds of pages like some books. It then has a few more chapters going into a bit more depth but on topics that are also essential if you want to take things further
- tamnd 1y ago[Author here] We hear you! Here's a similar book for Calculus: https://github.com/the-little-book-of/calculus https://github.com/the-little-book-of/calculus In this book, I cover Functions, Derivatives, Integrals, Multivariable Calculus, and Infinite Processes. In addition, I've included appendices with sketch proofs and applications to Physics, Probability and Statistics, and Computer Science.
- piperly 1y agoIt is not visible to the public, I think.
- jonahx 1y agoFor an unconventional but beautiful presentation of the subject, I highly recommend "Wild Linear Algebra" by the brilliant Norman Wildberger: https://www.youtube.com/watch?v=yAb12PWrhV0&list=PLBQcPIGljHUCo00lvKlQaxUgJ8s-Ei9q8 https://www.youtube.com/watch?v=yAb12PWrhV0&list=PLBQcPIGljH... It starts with the axioms of being able to draw one line parallel to another, and a line through point, and builds up everything from there. No labeled Cartesian axes. Just primitive Euclidean objects in an affine space.
- deleted 1y ago[deleted]
- eftychis 1y agoMy recommendation: Linear Algebra Done Right by Axler: https://linear.axler.net/ https://linear.axler.net/ Starts from linear transformations and builds from there.
- bobajeff 1y agoI love this. It's a great companion to all the resources online like Kahn Academy, 3blue1brown etc. and mathisfun, Wolfram Mathworld and Google's Gemini. Edit: Also the mathematics stackexchange.
- ziedaniel1 1y agoA little bit too concise IMO. E.g. it doesn't really explain where the normal equations come from and it mentions eigenspaces without ever defining them.
- tamnd 1y ago[Author here] Linear Algebra is the first book in a series I'm writing on math, computer science, and programming. You can find the index for the upcoming books here: https://github.com/the-litte-book-of/everything https://github.com/the-litte-book-of/everything. If the style clicks with you or you're curious about certain topics, I’d love to hear your thoughts!
- marginatum 1y agoThe pdf does not open.