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...
An Intuitive Guide to Linear Algebra (2012)
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Re: An Intuitive Guide to Linear Algebra (2012)
#22Does something like this exist for differential equations?
Re: An Intuitive Guide to Linear Algebra (2012)
#23This 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
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?
Re: An Intuitive Guide to Linear Algebra (2012)
#24This 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
> 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.
Re: An Intuitive Guide to Linear Algebra (2012)
#25I 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,…
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.
Re: An Intuitive Guide to Linear Algebra (2012)
#26This 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
> 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…
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)
Re: An Intuitive Guide to Linear Algebra (2012)
#27Earlier quoted context omitted.
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 r…
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
Re: An Intuitive Guide to Linear Algebra (2012)
#28Earlier quoted context omitted.
> 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…
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'…
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 lot of non-overlap, of which your list covers some. This article should be renamed to be about matrices and not linear algebra, because it's not.
Re: An Intuitive Guide to Linear Algebra (2012)
#29Earlier quoted context omitted.
> 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…
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.
Re: An Intuitive Guide to Linear Algebra (2012)
#30I 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,…
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 r…
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.).