Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

An Intuitive Guide to Linear Algebra (2012)

betterexplained.com

Post reply on HN
Print view View original

91–100 of 105 posts

Re: An Intuitive Guide to Linear Algebra (2012)

#91

Post by Chris2048 »

robpal wrote: post #76
id_ris wrote: post #61

I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. Now that the ideas of things like vector spaces, norms,…

My experience from a decade of doing professional maths is that there are no shortcuts. You only learn maths by doing hundreds (thousands) of exercises, both mundane and more exctiting. Also, the concepts "mature" in the brain. I remember sleepless nights in the first year of undergrad spent on understanding the details of the proof of the Jordan decomposition and a few years later (when studying algebraic groups) it…

I think there is a shortcut: use examples (a "domain") rather than starting with theory. learning something purely in the abstract is easy to forget.

Re: An Intuitive Guide to Linear Algebra (2012)

#92

Post by bakuninsbart »

id_ris wrote: post #61

I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. Now that the ideas of things like vector spaces, norms,…

I agree, and have to admit that my own knowledge of LA is sadly way too superficial. The article did give a me a big lightbulb on something I didn't understand before: Some of my friends work in quantitative sociology and economics, and use Stata for their programming needs. They basically use matrices as their exclusive data structure, and the reason for that eluded me until after reading this article.

Re: An Intuitive Guide to Linear Algebra (2012)

#93

Post by id_ris »

dfabulich wrote: post #68
id_ris wrote: post #61

I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. Now that the ideas of things like vector spaces, norms,…

> While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. I'm not sure I understand your point. Are you just saying that this blog post isn't an adequate substitute for taking a course in linear algebra? (Of course it isn't. But who said it was?)

There is a lot of excitement around machine learning and AI and there is not a proportional excitement for the math that underlies the theory.

In a lot of content that teaches machine learning/AI, the linear algebra substrate of it is given short shrift.

The consumers of that content infer that the backing mathematics is easy or unimportant, while the creators of the content are not actually implying that, but just want to move on to what the audience came for.

I'm not an expert in machine learning so I can't say whether you can get by without a strong understanding of linear algebra, but my intuition answer is no you can't. Beside that, I enjoy the math for its own sake so I'm happy to trudge through the textbooks.

And no, I don't think the blog post makes any claims that it is a subsitute for taking a course in linear algebra.

Re: An Intuitive Guide to Linear Algebra (2012)

#94

Post by FabHK »

smallnamespace wrote: post #53
xscott wrote: post #26

Earlier quoted context omitted.

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'…

A covariance matrix naturally transforms from the measured space to a space where things are approximately unit Gaussian distributed. This is identical to the Z transform in 1D case. This can be useful in, say, exotic options trading - a natural unit of measurement is how many ‘vols’ an underlier has moved, e.g. a 10-vol move is very large.

Not really the covariance matrix, though, but its Cholesky decomposition (which exists, as a covariance matrix is symmetric positive (semi)definite, as otherwise you could construct a linear combination with negative variance). Useful stuff.

And vice versa, btw - take iid RV with unit variance, hit them with the Cholesky decomposition, and you have the desired covariance. Used all over Monte Carlo and finance and so on.

Re: An Intuitive Guide to Linear Algebra (2012)

#95

Post by FabHK »

JadeNB wrote: post #29
xscott wrote: post #26

Earlier quoted context omitted.

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'…

> - A basis set of column vectors Let's leave the word 'basis' out, since the column vectors may well be linearly dependent.

Or not span the space, for that matter.

Re: An Intuitive Guide to Linear Algebra (2012)

#96

Post by FabHK »

trevor-e wrote: post #66
adamcharnock wrote: post #36

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…

Personally I really struggle with the syntax and notation of upper-level math courses, I need a big cheat sheet of all the terms. It's like programming where we use i, j, k for loop variables instead of something more descriptive. My brain has to do one extra layer of translation between what's written and the concept being taught and then I lose focus, but intuitively I've always been fine with math concepts. When I…

That is to a large extent a matter of practice. The notation can become second nature and then really helpful - you can write down regression (ordinary least squares) in two lines, basically (including problem and solution), and it all makes sense and translates fairly directly into an algorithm.

Re: An Intuitive Guide to Linear Algebra (2012)

#97

Post by mattkrause »

abdullahkhalids wrote: post #78
dfabulich wrote: post #68

Earlier quoted context omitted.

> While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. I'm not sure I understand your point. Are you just saying that this blog post isn't an adequate substitute for taking a course in linear algebra? (Of course it isn't. But who said it was?)

Unfortunately, for a lot of people, including undergraduates, the dopamine hit they get from watching a video or passively reading a textbook makes them believe that these are adequate substitutes for doing thousands of exercises. In my university, undergraduates have admitted that they have done fewer than 50 questions throughout the entirety of my math course. Their grades obviously reflect that, but they will do t…

Can I press you to explain what you think dopamine is/does?

I'm curious because I'm a neuroscientist who occasionally works on dopaminergic stuff and "passively reading a textbook" is so far from the canonical examples we use for dopamine activity, but the idea of dopamine/dopamine 'hits' has taken on a life of its own that seems quite different from the neurotransmitter.

Re: An Intuitive Guide to Linear Algebra (2012)

#98

Post by FabHK »

Swizec wrote: post #27
jimhefferon wrote: post #17

Earlier quoted context omitted.

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…

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

Corollary: a good teacher hasn’t understood anything. Well, that’s wrong, I think.

However, this might be true: A good teacher remembers not having understood it.

Re: An Intuitive Guide to Linear Algebra (2012)

#99

Post by mindcrime »

obastani wrote: post #49
adamcharnock wrote: post #36

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…

Part of the challenge with linear algebra is that a lot of the basics are somewhat dry, since they serve mostly as a way to organize computation -- e.g., a system of linear equations can be expressed as Ax Much of what makes linear algebra interesting and powerful comes from more advanced topics, especially eigenvalues. This power comes when we are not looking at a single matrix in isolation, but when we repeatedly a…

This post really illustrates why HN needs MathJax support, and/or browsers need MathML support. Math notation is hard enough to read when you're reading the real version... when you're forced to read someone's ad-hoc "notation that I'm forced to use because I can't use real notation" it just becomes that much worse.

Luckily, Chromium is getting MathML support (finally) unless something weird happens, which (presumably) means that Chrome and Edge will inherit that support as well. Firefox already has MathML, so hopefully we'll soon be in a position where three of the most widely used browsers support MathML.

Still, it would be interesting to ask the Powers that Be if they'd be willing to implement MathJax here on HN....

Re: An Intuitive Guide to Linear Algebra (2012)

#100

Post by JadeNB »

FabHK wrote: post #95
JadeNB wrote: post #29

Earlier quoted context omitted.

> - A basis set of column vectors Let's leave the word 'basis' out, since the column vectors may well be linearly dependent.

Or not span the space, for that matter.

Well, it depends on what "the space" is. Every set of vectors (in a common ambient space) spans some space—often called the column space of a matrix, if they are the column vectors of the matrix.
Post reply on HN