[HN Gopher] An Intuitive Guide to Linear Algebra (2012)
___________________________________________________________________
An Intuitive Guide to Linear Algebra (2012)
Author : PNWChris
Score : 260 points
Date : 2020-02-25 19:19 UTC (3 hours ago)
(HTM) web link (betterexplained.com)
(TXT) w3m dump (betterexplained.com)
| sushisource wrote:
| I love explainers like this, but it frankly makes me a little
| angry that the vast majority of the math teachers I had in
| highschool and college taught in the awful way described in the
| setup to the piece.
|
| Why is that? Has anyone studied it, or is there even a solid
| anecdotal explanation? The best one I can imagine is many of
| these professors simply don't care much for teaching and are more
| focused on their research, which is still infuriating but at
| least an explanation.
|
| I ended up with an undergrad in applied math, though I'm a
| software engineer now. I like math, but I feel like I never got
| to be all that great at it. I suspect I would've enjoyed it more
| and achieved more with explanations like these.
| salty_biscuits wrote:
| Here are two reasons I think are important, there are more but
| these stand out. (1) Teaching is hard, and the amount of
| training you get in teaching higher level maths (rather than
| the just learning higher level maths itself) is very limited.
| (2) intuition in maths will lead you on a merry path to very
| wrong ideas. There are functions that are continuous everywhere
| without a derivative anywhere, Russell's paradox, classically
| zenos paradox, etc, etc. A big part of higher level maths
| training is in being precise with what you mean so that you can
| learn to do proofs and not trick yourself. I think a synthesis
| of these approaches is the sweet spot. Be precise but show
| applications to motivate the material.
| mhh__ wrote:
| Have you ever taught?
|
| I taught Astronomy to the people in the years below me (a
| school tradition because it was optional), and it was
| absolutely exhausting trying to plan _good_ lessons that didn
| 't involve getting them to memorise stuff by wrote. I
| ""derived"" Kepler's laws for a bunch of 14-15 year olds who
| didn't even know logarithms yet and it was pretty brutal
| intellectually.
|
| Also, I think teaching mathematics "intuitively" requires a bit
| of cooperation from the student - not in the sense of
| intelligence, just that for for every guy (or girl) who watches
| 3Blue1Brown (and looks deeper into the pure mathematics)
| there's another who's just along for the ride. I think the
| frequencies of those personalities are a product of how they
| were taught, but it's very difficult to convince people in
| "teaching time" as opposed to naturally (I was in the lowest
| maths group for years until I picked up a calculus book on a
| whim, but no teacher could've convinced old me that mathematics
| can be beautiful)
| addicted wrote:
| How much of this explainer however seems better to us precisely
| because of the more comprehensive knowledge and understanding
| we already have?
|
| For example the author uses the word function liberally in the
| explanation. However, when studying functions in school in math
| it was super complicated for me. It was only after I started
| programming, learning the programming language meaning of
| function, and then when I was reintroduced to mathematical
| functions through functional programming that I truly grasped
| mathematical functions and all the stuff I was taught in
| school.
|
| I'm not arguing that school level math does not need to be
| improved. I just think we should be cautious that because an
| explanation that seems intuitive to us now after having gained
| a complete introduction to all math concepts as well as
| programming (esp on HN) may not necessarily be as intuitive
| when students who are only exposed to a very small subset of
| mathematical concepts encounter them.
| swiley wrote:
| As someone who learned to program before taking any
| interesting math I was always so confused by the amount of
| early math classes spent explaining function notation.
|
| It wasn't until I tried teaching people that I realized how
| odd the notation can seem.
| kevstev wrote:
| you should be happy someone explained it to you- I feel
| that notation was overlooked tremendously in my math
| education- I actually just brought this up in a thread
| yesterday. It was like "oh, we have this dx now... k."
| jimhefferon wrote:
| For sure there is an effect on Reddit and other places where
| someone will post a question such as, "I'm having trouble
| with my Calc class, what is a good book?" and people
| seriously answer _Calculus on Manifolds_.
|
| Now, CoM is a classic, a real great book, but it is useful
| only to people who have reached a certain level of
| mathematical maturity. That, presumably, is not the
| questioner.
|
| A version of this is that I also see people who write, "I
| didn't understand this topic when I took the course but now
| years later, I see it is all actually very easy."
|
| (I call this Second Book Syndrome because I don't know of a
| common name for it. I understand this to be what Zen people
| mean by the "Gateless Gate," that after struggling with
| something at great length a person can come to see that there
| is no real difficulty. But I've never heard anyone else apply
| that name to this phenomenon and I'm not Zen trained so I'm
| not sure that is right either.)
| Swizec wrote:
| Once you understand monads, you lose the ability to explain
| monads. Hence the number of monads tutorials grows at an
| exponential rate as every new understander tries to explain
| them and fails.
|
| it's a fun problem in teaching
| jfarmer wrote:
| Ironically, I think the original paper that introduced
| Monads as a useful computational abstraction is the
| clearest explanation I've seen.
|
| https://homepages.inf.ed.ac.uk/wadler/papers/marktoberdor
| f/b...
| Koshkin wrote:
| > _Once you understand... you lose the ability to
| explain_
|
| Sorry, this does not make sense to me.
| cycomanic wrote:
| Actually a rather famous example of your point are the
| Feynman lectures. They are often hailed as a great, "easy"
| and intuitive explanation of physics. However, supposedly at
| the time, the undergrad students Feyman was teaching did not
| think their were all the rage. It was actually the grad
| students who retook the material and really enjoyed them. To
| support your point I think sometimes it is really necessary
| to have been exposed to the material and been able to grasp
| some part of it, then you are open to these type of
| explanations which really lift you over the hurdle of the
| things that you didn't quite get before.
| just_myles wrote:
| I agree with your post. I had the same problem where match
| was harder for me to comprehend. It wasn't until I started
| programming that I started to understand the concept you cite
| (Functions.).
| originalvichy wrote:
| I think the problem might be that the kids best at math become
| teachers. The ones that it "came to easily".
|
| This creates the not-so-obvious problem that the easier you
| learn something the harder it might be to teach it to someone
| who doesn't get it.
|
| If you on the other hand struggle hard to find a way to
| understand a complex concept, you might be good at transfering
| that knowledge forward.
| bgroat wrote:
| I love betterexplained.
|
| Does this guy have a patreon?
| wodenokoto wrote:
| He has a business that sells courses and books right on the
| homepage.
| RickS wrote:
| Already this is so helpful, as someone with only a partial high
| school math background. The visual "pouring" analogy alone made
| it worth the read.
| adamcharnock wrote:
| I... I still really struggle with this. I'm a smart person, I've
| got a bachelors of engineering, I've been a professional software
| developer for around 14 years now, and I've built a house. But
| there is something about degree-level maths and beyond that I
| find deeply unintuitive in a way that software development isn't.
|
| Through comments here I found 3blue1brown's (clearly much loved)
| videos. By the third video I was shouting, "why for the love of
| god would we be doing this"? Based on this reaction I suspect
| that the content neither has intrinsic appeal to me, nor does it
| have obvious use in my work, projects, or life.
|
| Pre-degree maths though, I love. My A-level maths really changed
| how I saw the world, and I make use of it reasonably often (well,
| often enough to not forget it).
|
| I think I'm writing this here because most other commenters seem
| to really grasp this subject, or feel that they grasp it better
| having seen these videos. I'm honestly happy for you. However, if
| anyone is reading this who doesn't feel like that, then know
| you're not alone :-)
| woah wrote:
| I used to think this, my problem was that I wasn't doing the
| exercises, instead just reading articles and watching videos
| and trying to get some kind of theoretical understanding.
| Everything makes a lot more sense once you've slogged through a
| bunch of repeated exercises.
| amelius wrote:
| Try Gilbert Strang's course at MIT, it's publicly available in
| video format, and starts from basic principles.
| dragontamer wrote:
| The problem with Linear Algebra specifically, is that it can be
| viewed from many different perspectives.
|
| The article here focuses on an "operational" perspective, how
| the numbers get added or multiplied together to turn into other
| numbers. However, Linear Algebra is also useful in geometry,
| and other situations.
|
| This "intuitive guide" to linear algebra sets you up very
| nicely for figuring out how to add and multiply matricies
| together. But it doesn't give you any intuition about a
| rotation (aka quaternions) in 3d space, for example. A lot of
| math books make the mistake of trying to teach all the
| perspectives at the same time, instead of focusing on just one
| viewpoint until the student gains mastery.
|
| A Quaternion is "just" a 4x4 matrix that represents rotation in
| 3-dimension space. Because you only move 3-ways rotationally
| (yaw, pitch, and roll), you're "underconstrained" with regards
| to the 4x4 matrix. Etc. etc. A lot of geometry intuition needs
| to be built here to really understand Quaternion... and none of
| that geometry is explored in the blogpost.
|
| Which is fine. Focus is good. But when people approach Linear
| Algebra, its important to know that its "so useful" that there
| are too many ways of looking at Linear Algebra... too many
| different, yet equivalent, understandings of the subject.
| hintymad wrote:
| This is typical for many people. You love pre-college math
| because you have intuitive understanding, while college-level
| maths offer a new level of abstraction that you may not feel
| familiar with from the get-go.
|
| It's perfectly okay not to learn linear algebra, by the way,
| especially when you don't find any incentive to do so.
| Otherwise, you'll find linear algebra to be one of the most
| intuitive tools to model so many problems.
|
| If you do want to learn linear algebra or any other higher
| math, I'd strongly recommend you focus on understanding
| concepts intuitively first, to the point that you find many
| exercises in a text book straight forward. Watching 3blue1brown
| is a good start, but do move forward with deeper treatment. The
| book I find very usual is David Lay's Linear Algebra and Its
| Applications: https://www.amazon.com/Linear-Algebra-Its-
| Applications-5th/d.... Lay sets up a really intuitive geometric
| framework to explain the intuition of linear transformation
| with sufficient rigor.
| vecter wrote:
| This is ok but nothing is as intuitive as 3B1B's series on
| YouTube that has been posted hundreds of times on HN [0].
|
| Linear algebra is really about linear transformations of vector
| spaces, which is not captured in this blog post.
|
| [0] https://www.youtube.com/watch?v=fNk_zzaMoSs
| foxx-boxx wrote:
| Often these articles are written by students who either failed
| their exam or scared to fail it. People who actually know the
| subject get paid either for lectures or for real work.
| sandov wrote:
| Playlist link:
| https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
| OrwellianChild wrote:
| 3B1B does _great_ work explaining these concepts, but I can 't
| help but ask "why not both?" when it comes to explaining these
| concepts. Turns out, linear algebra is great for working with
| matrices, vector space, approximating non-linear systems, and
| more... Let's embrace multiple ways of teaching it and gaining
| intuitions rather than keeping score, eh?
| nightcracker wrote:
| > Linear algebra is really about linear transformations of
| vector spaces, which is not captured in this blog post.
|
| I... disagree. Some of linear algebra is about that. And it's
| probably a good way to view it that way when learning.
|
| But some of my current work (coding theory) involves linear
| algebra over finite fields. We use results from linear algebra,
| and interpret our problem using matrices, but really at no
| point are we viewing what we're doing as transforming a vector
| space, we're just solving equations with unknowns.
| xscott wrote:
| I think this is spot on. Depending on what you're doing, a
| matrix can be: - A linear transformation
| - A basis set of column vectors - A set of equations
| (rows) to be solved - (your example: parity
| equations for coding theory) - The covariance of
| elements in a vector space - The Hessian of a
| function for numerical optimization - The adjacency
| representation of a graph - Just a 2D image
| (compression algorithms) ... (I'm sure there are
| plenty of others)
|
| For some of these, the matrix is really just a high
| dimensional number. You (rarely?) never think of covariance
| in a Kalman filter as a linear transform, but you still need
| to take its Eigen vectors if you want to draw ellipses.
| vecter wrote:
| Great points. I wrote my comment in response to the article
| claiming to be an intuitive guide to _linear algebra_ , not
| an intuitive guide to matrices. According to wikipedia:
|
| > Linear algebra is the branch of mathematics concerning
| linear equations, linear functions, and their
| representations in vector spaces through matrices. [0]
|
| The Venn Diagram of Linear Algebra and Matrices definitely
| has a not of non-overlap, which your list covers some of.
| This article should be renamed to be about matrices and not
| linear algebra, because it's not.
|
| [0] https://en.wikipedia.org/wiki/Linear_algebra
| JadeNB wrote:
| > - A basis set of column vectors
|
| Let's leave the word 'basis' out, since the column vectors
| may well be linearly dependent.
| threatofrain wrote:
| How I view it is the matrix is a 2-dimensional indexed data
| structure, but when conditions are right the matrix becomes
| an object of Linear Algebra.
| snicker7 wrote:
| Linear algebra is a shared field across multiple disciplines.
| So I'm sure that there are many valid and useful
| interpretations as to what "linear algebra" is essentially
| about.
|
| However, in mathematics proper, it is absolutely the case
| that linear algebra is about linear transformations. Indeed,
| this is the only interpretation that remains meaningful when
| trying to generalize (e.g. to functional analysis /
| multilinear algebra).
| adamnemecek wrote:
| If this interests you, you should check out the bivector
| community https://bivector.net/.
|
| Join the discord https://discord.gg/vGY6pPk.
|
| Check out a demo https://observablehq.com/@enkimute/animated-
| orbits
|
| Also at the end of February, there is geometric algebra event in
| Belgium. https://bivector.net/game2020.html All the big names in
| the field will be there.
| nearlynameless wrote:
| While this explanation is certainly much clearer than what I
| remember of high school maths, I still have a pretty tough time
| following the formula examples.
|
| When I see A(x) = ax, I'm not entirely sure how to read it.
|
| Is A meant to be a function that accepts x? If so, why is the
| equivalent expression a * x? Is it supposed to be implied that
| function A also has some hidden value "a" that is going to be
| multiplied by the supplied value? Is this notation specific to
| multiplication, to this expression, or what?
|
| Positing that something is 'intuitive' when it depends so much on
| additional contextual knowledge seems ever so slightly
| disingenuous as best, and slightly harmful at worst; it can make
| the reader feel as though they must be dumb for not understanding
| this 'intuitive' material.
|
| I do acknowledge that this is linear algebra, and if one doesn't
| have a really solid grasp of notation of regular algebra it is
| likely to go over their heads, but the practical explanations
| (such as the slope rise/run example) are quite clear and
| relatively simple to follow; it follows that a simple explanation
| of the notation might be helpful too.
| kdtop wrote:
| I agree that this was a bit confusing. Higher up in the
| article, it shows that a linear function is one that doesn't
| change when scaled:
|
| F(a * x) = a * F(x)
|
| This is showing the relationship between two uses of the same
| function.
|
| Then, further along, we find:
|
| "So, what types of functions are actually linear? Plain-old
| scaling by a constant, or functions that look like: F(x)=ax In
| our roof example, a=1/3"
|
| I think in this second situation, F(x)=ax is not a relationship
| but rather a DEFINITION of the function F(x).
|
| In programming terms:
|
| function F(x: real) : real;
|
| begin Result := x * (1/3);
|
| end;
| the_watcher wrote:
| I remember taking Advanced Algebra Honors in 10th grade. It was
| basically Algebra II with a few (seemingly teacher-selected based
| on the experience of students who had a different teacher)
| advanced topics thrown in. One of them was matrices, and I was
| completely stumped by them. I now encounter them all the time,
| and wish I'd been able to wrap my head around them when I was
| younger.
| marcuskaz wrote:
| 20+ years ago I combined learning JavaScript and Linear Algebra
| https://mkaz.blog/math/javascript-linear-algebra-calculator/
| alacombe wrote:
| Speaking for Linear Algebra, I learnt more reading for a few
| hours the appendix of "The Design of Rijndael: AES - The Advanced
| Encryption Standard" than I did in 6 months of theoretical
| university teaching full of useless technical terms and solutions
| in search of problems...
| JadeNB wrote:
| > solutions in search of problems...
|
| This sounds like it was meant to be pejorative, but it's what
| (applied) linear algebra, and applied mathematics more
| generally, _is_. Anyone can learn about a certain mathematical
| topic upon realising it 's the one relevant to the problem
| they're facing--and learn it way more quickly, due to
| motivation and focus, than they would in a general-purpose
| course on the topic; the art is in recognising what mathematics
| is relevant, and you can't do that if you've never heard of it
| before. Having a library of (conceptual, not cookbook) solution
| hooks on which to hang your problems is how you get to be good
| at using mathematics.
| PNWChris wrote:
| OP Here:
|
| I just wanted to give some context to how I found this page, and
| why I thought it would be good to post.
|
| I may be putting myself on the spot here: I never took a linear
| algebra course in undergrad. It was a heavily encouraged option,
| of course, but I felt I understood the basic rules enough to not
| really need formal study. I opted to study other areas, partially
| motivated by a fear I wouldn't do well and would hurt my GPA (my
| god was I vain, I feel I could do so much more studying full-time
| with my current world-view).
|
| As time has gone on, and ML and quantum computing have simply
| blown up since I graduated in 2014, I quickly realized the
| magnitude of my mistake. I have frantically self-studied for
| years to try to make up the gaps in my mathematical
| understanding, and linear algebra has come up time and time
| again. I can do the processes, but they never clicked, I had no
| intuition.
|
| I want to help others in my position cut to the chase, and study
| the highest yield, most intuition giving resources.
|
| I actually developed the mental model shared in this guide on my
| own, and was positively delighted to find to this while thinking
| over a comment I was drafting on here. This page lays things out
| so clearly. The component steps are intuitive and I can commit
| them to memory/recall what they mean without needing to dig up my
| notes to self!
|
| ===
|
| I find this page gives an excellent foundation, and goes great
| with these resources:
|
| * A web site which clearly shows how to do matrix multiplication
| in a way that's easy to recall, it makes the procedure like
| riding a bike:
|
| http://matrixmultiplication.xyz/
|
| _Huge thanks to Jeremy Howard of fast.ai for mentioning this in
| one of his lectures, this tool is how I finally got matrix
| multiplication to click_
|
| * A paper named "An Introduction to Quantum Computing" (bear with
| me, it's superbly well written and very approachable):
|
| https://arxiv.org/abs/0708.0261
|
| Page 3 of that paper lays out matrix multiplication (e.g.:
| applying a "transformation matrix" in the spatial parlance of
| 3blue1brown's videos) as a traversal of a directed graph. A very
| useful understanding, and shows how generalizable the tools of
| linear algebra really are in my opinion.
|
| * The essence of linear algebra, by 3blue1brown (fantastic for a
| geometric/"transformation of space" view of linear alg):
|
| https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
| dang wrote:
| A thread from 2015: https://news.ycombinator.com/item?id=8920638
|
| Discussed at the time:
| https://news.ycombinator.com/item?id=4633662
| dundercoder wrote:
| Does something like this exist for differential equations?
| formalsystem wrote:
| Viewing Linear Algebra as the study of linear operators instead
| of matrices makes everything so much simpler.
|
| Of course AB != BA
|
| Composition makes sense
|
| Inverse makes sense
|
| This is the book that helped me get it http://linear.axler.net/
| mathnovice wrote:
| I'm reading Strang's linear algebra book and he teaches it in
| terms of combining columns and rows which I think is a lot
| clearer than explaining it in terms of linear equations.
| swiley wrote:
| I really like "matrices are the coordinate form of linear
| transforms." In the LA class I took the professor made a pretty
| big deal out of that, first by defining "lexicographic matrix
| basis" so he could write out matrices as vectors and then
| talking about mapping between the three different ideas.
|
| There's still stuff he said that I'm unpacking today... that
| was a dense class.
| zelly wrote:
| This is a great book. I also recommend Halmos "Finite
| Dimensional Vector Spaces". The typical way linear algebra is
| introduced does not present a matrix as a linear transformation
| first and foremost.
___________________________________________________________________
(page generated 2020-02-25 23:00 UTC)