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