Very beautiful graphs, but i don't think it's going to make people understand anything.
I would start with the problem : easily compute sums of dependant values, then show a naive computation, then use matrices, vectors and eigenvalues to come to a solution, and only then, show a graphical representation of the steps performed.
I'm surprised that this post isn't following this method, because i've come to think it's the standard way of explaining scientific things in the US.
Eigenvalues and eigenvectors are one of those things that pop up in a million places because they're so useful, but to recognize where they may be useful you need intuition as to what they're doing.
One of my biggest hurdles learning linear algebra was getting that intuition. This "standard way of explaining scientific things" never built that intuition for me, only forced the mechanics of the computation into pencilized muscle memory.
Don't get me wrong, you need that muscle memory in practice. But without the intuition, your muscle memory is always going to be inferior to a couple commands in matlab. Stuff like this visualization builds the intuitive knowledge -- you see, you explore, you wiggle a few things and see what happens when you go in and out of the sweet spot.
Play (and simulation) is an extremely effective way to build intuition, and one I'd love to see more. These guys are doing an awesome job at these kinds of simulations -- their markov chain one was fantastic too http://setosa.io/ev/markov-chains/
>Stuff like this visualization builds the intuitive knowledge -- you see, you explore, you wiggle a few things and see what happens when you go in and out of the sweet spot.
Disagree. The linked article does a very poor job of explaining anything, much less conveying intuition. The lambda values remain even if the 3 points are not collinear, thus contradicting the first part of the article.
From what you've said i think you mean that it isn't the first time you've tried to understand them, or be exposed to their definition. In which case yes, absolutely the graph representation of base change exposed here is very helpful.
I was more talking for people that would start with this page.
What i meant was that the fact that the page starts right away with base change, and introduces practical uses of it later seems akward.
Insanity. 90% of linear algebra can be understood from graphically examining affine transforms. Doing visuals last is what leads people to conceive of matrix-vector multiplication as a set of meaningless dot products as opposed to just multiplying the basis vectors by the matching coordinate and adding them all up.
I think you're missing the first step, which is, formalizing a problem using matrices and vectors in the first place.
Correct me if i'm wrong, but i can easily imagine that people have been solving those types of issues manualy for centuries before finding the "shortcut" or representing transforms using matrices and figuring the rules of matrix & vector multiplication.
Still, i think you're making a good point. So maybe the correct process would be : problem stating, naive "manual" solution, graphical representation and then matrice & vector formalizing ?
I'm surprised that this post isn't following this method, because i've come to think it's the standard way of explaining scientific things in the US.