The point then is that the taps (again, representing the matrix values) determine how much of each item in the input vector, that should be mixed into each item in the output vector.
This analogy has the limitation that the taps are allowed to enhance the flow, not just limit it, like physical taps would. That is, outputting more than 100% of the input :P Also, while this way of illustrating it may make some sense for matrix * vector multiplication, matrix * matrix would probably become a prohibitively cluttered image.
Look at the animation in the article, after the second matrix has been rotated and put on top of the first one. Then flip its top up so the two matrices are orthogonal. Finally, rotate the whole thing 90 degrees to the left (rotating along the axis that goes from the top of the page to the bottom of the page). Now you can see that the result will be a 2x3 matrix. And the values in each spot will be the sum of the products beneath it.
He has a whole series on linear algebra, including matrix mulitplication:
https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
I actually disagree with Axler on his avoidance of the determinant, though. I wish instead of avoiding it he'd spent more time developing it conceptually, as it's actually a fascinating construction. But to this day I have yet to find a gentler and better introduction to serious linear algebra than his book.
He recommends it as a second course, but I read it during my first course in the subject and considered it my "secret weapon". I truly believe that book is what allowed me to get a perfect score in the class -- I had a conceptual understanding that was just not possible to glean from the official course textbook.
Axler's book covers more ground (most notably, Halmos presents the polar decomposition but not the singular-value decompostion) and uses more modern terminology and notation. But Halmos's book has the merits of being half as long and a third as expensive, as well as having been written specifically to prepare the reader as directly as possible for Halmos's short introduction to Hilbert spaces [1].
I highly recommend one or the other of these books for readers who want to understand linear algebra as mathematicians do.
0. https://www.amazon.com/Finite-Dimensional-Vector-Spaces-Paul...
1. https://www.amazon.com/Introduction-Hilbert-Theory-Spectral-...
I could never learn mathematics that way, and goodness knows, school systems in both Europe and America have tried to teach it to me that way. What I understand of mathematics today has mostly been achieved through autodidaction.
Some of us cannot, repeat can not go from the abstract to the empirical; I'm one of those people, who have to go from the empirical to the abstract instead. I need to see the mechanics of it, because I learn by observing the pattern. If I cannot make out the pattern, I cannot comprehend the abstraction.
Take UNIX manual pages for example: the first thing I go to after reading the SYNOPSIS is the EXAMPLES section (and since most GNU/Linux manual pages have no EXAMPLES sections, it's a product which is useless to me, unlike UNIX). The more examples the EXAMPLES section entails, the faster I'm able to grasp the concept. Same with http://matrixmultiplication.xyz/, if I could have learned matrix multiplication that way, visually, instead of having to do it in my head, it would have been far easier. The other way around, it was a slow, painful torture, and to this present day, I cannot multiply the matrices in my head, or without these visualisations to remind me of the rules: they're too complicated for me to keep in my head.
The point I'm trying to make is that not everyone learns the same way, and our brains do not process the information in the same way, even if the final result, understanding of the concept, is identical. A lot of abstract concepts which are difficult to reliably repeat, or whose outcome is either non-deterministic or unclear, we do not even understand the same way, hence opinions often differ by a small or large margin. Don't assume that the learning technique which works for you would work for someone else. That's a major failure of most pedagogical approaches, with the exception of perhaps Montessori and Fröbel. Unlike Montessori and Fröbel, most pedagogical approaches force a uniform way of learning, except that a human brain does not learn in a uniform manner, differing from individual to individual instead. Teaching should be a highly individualized approach.
This is actually pretty instructive. For instance, rotations in three-dimensional space do not commute in general; hence neither does matrix multiplication.
Once I had finished that, Cullen's "Matrices and linear transformations"[2] was really helpful too. But I wouldn't do Cullen if you're still, as I was, floundering with the concepts of why you're doing this in the first place. It's great once you have those concepts down.
[1]: https://www.amazon.com/All-Mathematics-You-Missed-Graduate/d...
Now let's see what happens after two steps. If the system started out in state i, what's the probability that after two steps it will end up in state k? Well, it's the sum over all possible paths. In other words, the sum of probabilities of i->j->k for all possible j. In other words, the sum of p_{ij} times p_{jk} for j from 1 to N. But that's exactly the definition of multiplying a matrix by itself.
Now it should be easy to understand that whenever you have matrices that represent transformations of some object, composing transformations will correspond to multiplying matrices.
[1]: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
http://math.stackexchange.com/questions/31725/intuition-behi...
Relying to much on explicit coordinates many times obfuscates what's going on.
It's strange. Somehow I have now problems whatsoever regarding the use of 4-component vectors. As long as the z component stays one, I'm all set. But lately I'm trying to understand the construction of projection matrices and how to apply the knowledge of homogeneous coordinates, for example shadow mapping.
Parallel lines suddenly intersect? 3D space is just a projective plane? Now everything is all weird...
[0] http://worrydream.com/ExplorableExplanations/