3Blue1Brown
6 HN recommenders · recommended in 7 months, 2019–2024
10:03 · 5M views · Aug 10, 2016
A 10-minute 3Blue1Brown chapter explaining the determinant as the factor by which a linear transformation scales area or volume. It builds the idea geometrically with animations before any formulas.
HN repeatedly points people to it as the clearest visual intuition for determinants, often saying it finally made the concept click.
7 comments from Hacker News
As someone who very much relates to the GP’s anecdote, I might suggest determinants as a good example. As an undergraduate studying maths, I encountered a standard theorem in one of my first courses, which says that about 947 different conditions are equivalent to a matrix having a determinant of zero. I dutifully memorised these. I also dutifully memorised the algorithm for how to calculate a determinant. I might even have remembered some verbatim proofs of some of the equivalences. However, I developed absolutely no intuition about what a determinant is . I had book knowledge, but no insight. It was a long time ago now, but I’m fairly sure that when I graduated I still did not truly understand even this very basic (by undergraduate standards) subject. I think it was probably a few years later, when I came across some of the same theory but in a much more practical context at work, that most of the connections in that equivalence theorem first “clicked”. Meanwhile, here is what a gifted presenter with the right illustrations can do in about ten minutes: this video The 2,000 or so substantially identical comments below that video are very telling. Given the understanding you’d get with that quality of presentation, the equivalences I mentioned above would have been obvious and constructing the proofs from first principles would have been straightforward.
At the same time, universities could sometimes do a better job of letting students see why it's important to learn something. A great example is matrix determinant, which at first glance seems like an entirely arbitrary invention. The 3Blue1Brown YouTube channel does a great job with this, introducing the determinant in terms of geometry. It puts the geometric meaning of the function first, introducing the strange looking formula later on. edit And of course, it does all of this with excellent visualisations. this video
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] [LINK] [1] this video
Determinants always fascinated me, partly (completely?) because of the cool sounding name. I always wanted to get intuition for them because their formula is so simple, which has always been deeply unsatisfying to me to just take on its face. This article got me searching for a 3blue1brown video on determinants and now my mind is absolutely blown! this video
To add to this, a website I've found useful is betterexplained.com. Specifically for complex numbers, take a look at [1] and [2]. But they also have good articles with visualizations to help you understand the Fourier transform, the number e, integrals and derivatives, linear algebra, etc. A starting point is [3]. One useful idea is that they color the different parts of math equations to explain each part separately, see [4]. Another great source of math visualizations is the YouTube channel 3Blue1Brown, in particular the series on linear algebra [5]. (I didn't intuitively understand what a determinant represented or what the determinant of a transformation being zero really meant, until watching [6].) [1] [LINK] [2] [LINK] [3] [LINK] [4] [LINK] [5] [LINK] [6] this video