13.1M views · Aug 6, 2016 · 9:52 · 17 HN recommenders · 20 mentions · discussed 2018–2026
The first chapter of 3Blue1Brown's Essence of Linear Algebra series, introducing vectors with animated geometric visuals. It begins a series that builds intuition for linear algebra from the basics.
Why Hacker News points people here: HN's go-to recommendation for learning linear algebra intuition, repeatedly credited with making concepts click and complementing textbooks.
“the most welcoming and informative introduction to the topic I've ever seen”HN comment, 2018
“the most welcoming and informative introduction to the topic I've ever seen”
Most welcoming and informative intro to linear algebra
The 3B1B series on Linear Algebra is by far the most welcoming and informative introduction to the topic I've ever seen: this video
“His explanations are so clear that by the 2nd or 3rd video you'll already understand how it applies to computer graphics.”
Clear explanations applying linear algebra to graphics
3Blue1Brown has a great series on Linear Algebra. His explanations are so clear that by the 2nd or 3rd video you'll already understand how it applies to computer graphics. this video
“is the go to place for learning this material”
The go-to series for learning linear algebra
3Blue1Brown's Essence of linear algebra series is the go to place for learning this material. I've included the link to the series below so people don't get lost in all his other wonderful videos along the way. this video
“3blue1brown's playlist on the subject is just ... perfect.”
Perfect visual intuition for linear algebra including eigenvectors
If you want to get an intuitive, visual understanding of linear algebra -- including eigenvectors/eigenvalues -- 3blue1brown's playlist on the subject is just ... perfect. this video
“nothing is as intuitive as 3B1B's series on YouTube”
Most intuitive linear algebra treatment, via linear transformations
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] this video
“I strongly recommend 3blue1brown's intro to linear algebra[0] for quick intuitions”
Quick intuitions, graphics helpful early on
I strongly recommend 3blue1brown's intro to linear algebra[0] for quick intuitions, followed by Gilbert Strangs MIT Open Course[1][2] for (much) deeper detail. Strang is a genius, but Sanderson's graphics and intuitions can be helpful early on. [0] this video [1] link [2] link
“The videos on linear algebra in particular are worth it, especially if you do any sort of machine learning”
Nicely made math explainers, linear algebra videos especially worth it
For anyone wondering what 3Blue1Brown is, it's a top Youtube channel with very nicely made explanatory videos for math. The videos on linear algebra in particular are worth it, especially if you do any sort of machine learning: this video . The author of the channel is Grant Sanderson. He has recently given a talk at SIGGRAPH 2021: link . As part of developing his Youtube channel, he has written and open-sourced manim, a library for programatically generating animations written in Python: link .
“Spending a few hours on this series will easily save you dozens of hours when going through a comprehensive LA textbook.”
A few hours saves dozens when studying a textbook
Here's another vote for 3Blue1Brown's series on linear algebra [1]. Spending a few hours on this series will easily save you dozens of hours when going through a comprehensive LA textbook. [1] this video
“Essence of linear algebra is an absolutely wonderful series.”
Gave intuition in hours that university years did not
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. this video
“3blue1brown's visualizations made a lot of things click for me”
Visualizations that made things click, a perfect companion to textbooks
3blue1brown's visualizations made a lot of things click for me, years after learning algebra and calculus. In my opinion they're the perfect companion to a more formal study with textbooks. Playlist for linear algebra: this video
“It is the Youtube channel 3Blue1Brown[1] that gave me an intuition of what linear algebra was about”
Gave intuition of what linear algebra is about
Linear Algebra Done right did not help me try to demystify linear algebra. Quite the contrary. It is the Youtube channel 3Blue1Brown[1] that gave me an intuition of what linear algebra was about, and I am forever thankful to him. Watch it and if you are a teacher, don't be a smug "formalist". [1] this video
“The entire 3blue1brown series[0] on linear algebra is well worth watching”
Really intuitive graphical explanations of linear algebra concepts
The entire 3blue1brown series[0] on linear algebra is well worth watching, it has really intuitive graphical explanations of a bunch of concepts. Here's the one on determinants in particular[1]. TL;DW the determinant represents how much you scale the area/volume/hypervolume (depending on dimension) of a shape by applying a matrix transformation to each point. [0] this video [1] link
“It has fantastic visualisations for building intuition and in general is wildly superior to this”
Visual intuition-building series on linear algebra, far better than the blog post
If you actually want to learn linear algebra, don't use this blogpost. It's real weaksauce compared to the wealth of free information and resources available online. Firstly, the real illustrated guide to linear algebra is the youtube series "The Essence of linear algebra" by 3blue1brown[1]. It has fantastic visualisations for building intuition and in general is wildly superior to this, which seems fine but extremely superficial. If you're done with 3b1b and want to take things further, then the go-to is the excellent 18.06SC course by the late and legendary Gilbert Strang. It's amazing, it's free. [2] Still want more? OK now you're talking my language. If you are serious about linear algebra (Up to graduate level, after that you need something else) then you want the book "Linear Algebra Done Right" by Sheldon Axler. It's available for free from the author's website[3] and he has made a bunch of videos to supplement the book. There's also an RTD Math full lecture series[4] that follows the book and he explains each thing in a lot of detail (because Axler goes fast , so it's beneficial to unpack the concepts a bit sometimes). [1] this video [2] link [3] link and link [4] link
“3Blue1Brown[1] has opened my eyes on the intuitive understanding of linear algebra”
Gave intuitive understanding of linear algebra after a confusing first course
3Blue1Brown[1] has opened my eyes on the intuitive understanding of linear algebra, many, many years after taking a first course in linear algebra, which left me completely mystified. [1] this video
“the amazing 3blue1brown "Essence of Linear Algebra" series”
Amazing visual Essence of Linear Algebra series
If you haven't already studied Linear Algebra, and want to get a headstart on that, check out the "Coding The Matrix" book/videos from Brown. link link link Also, see the Gilbert Strang video series on Linear Algebra: link and the amazing 3blue1brown "Essence of Linear Algebra" series: this video
“I think it's exactly that kind of "intuition of what Linear algebra is and is for".”
Builds intuition of what linear algebra is and is for
If you haven't seen it, it might be worth checking out 3Blue1Brown's Linear Algebra YouTube series ( this video ). I think it's exactly that kind of "intuition of what Linear algebra is and is for".
“It will give you an intuitive, geometric interpretation for concepts such as a vector, a basis, an eigenvalue, a linear transformation”
Intuitive geometric interpretation of vectors, bases, eigenvalues
Maybe you would get something out of 3blue1brown's series on linear algebra [1]. It will give you an intuitive, geometric interpretation for concepts such as a vector, a basis, an eigenvalue, a linear transformation, etc. As for why you would want to know these things, well, linear algebra is finding applications all over the place these days. Everything from computer graphics to machine learning is jam-packed with linear algebra. And I'm not just talking about basic concepts such as matrix-vector multiplication. The singular value decomposition has applications in discrete optimization, image compression, the PageRank algorithm [2], computer vision [3], and machine learning [2] [4]. Having said that, you may find it very difficult to understand something like SVD without a firm grounding in the topic of vector spaces, linear transformations, spanning, linear (in)dependence, subspaces, eigenvalues, eigenvectors, diagonalization, and determinants. This is why SVD is one of the last things you learn in a linear algebra course (indeed, it's not covered until lecture 29 of Gilbert Strang's course). Ultimately, it all depends on how relevant these things are to you. I would assume (hope) that because you clicked on this HN discussion that you're interested in learning linear algebra because you think it might be useful to you. [1] this video [2] link [3] link [4] link
“Those 2 video's will do a good job of giving you the 'intuition' behind Calculus and Linear Algebra.”
Intuition behind linear algebra and calculus
To answer your main question - "what level of maths" do you need is more about what topics: a couple introductory classes in Calculus, Linear Algebra, and Discrete Mathematics will make that sentence understandable. I'll give some definitions and resources, but think about how you would explain multiplication/division/algebra to a kid. At some point, you just have to work through some problems and the math just makes sense. So I'll explain the terms above, but best bet is to just take a few courses on the topics I mentioned above. Quick definitions: - Exponential: is the constant e to the power of some number. e, like pi, is a ubitiqous constant in mathematics and nature. You run into pi when doing geometry, and e when doing calculus. - A function in mathematics is similar to a function in programming, but not exactly. A course in Discrete Mathematics helps here. - F-apostrophe is notation used in calculus to show the relation of one type of function, called a derivative to another. f and f' are related. How, though, is better explained by taking calculus. - An eigenfunction exists in a system of equations. Think of it like a 'balance' point. Linear Algebra will help make sense of this term. Resources - Essence of Calculus: link - Essence of Linear Algebra: this video Those 2 video's will do a good job of giving you the 'intuition' behind Calculus and Linear Algebra. Like programming, though, you just have to actually work some problems out by hand for the stuff to sink in. Fo
“For intuition”
Recommended for intuition among several resources
There are many good resources! A few, depending on your inclination: - For intuition: this video - For rigor: link - For code: link - For numerical/algorithmic details: link
“You still have 3blue1brown[1] series of video, but it just brush off the surface.”
Series noted but said to only brush the surface
Frankly, couple with this book, it does hardly get better. You still have 3blue1brown[1] series of video, but it just brush off the surface. [1] this video
