This is very cool. For me I needed to know why it works not what it does.
I worked it out here for 2d transformations http://webglfundamentals.org/webgl/lessons/webgl-2d-matrices... and then expanded it to 3d, perspective, and cameras.
Matrix Multiplication
111–120 of 135 posts
Re: Matrix Multiplication
#112This is a cool example of what Bret Victor calls an "Explorable Explanation" [0]. That said, I feel that it's more important to understand how and why matrix multiplication corresponds to a composition of linear transformations than learning the actual mechanics of doing the computation. You can get good at matrix multiplication without knowing what is going on. I view that as a less valuable activity than learning a…
You'd probably like this then:
Re: Matrix Multiplication
#113This is a cool example of what Bret Victor calls an "Explorable Explanation" [0]. That said, I feel that it's more important to understand how and why matrix multiplication corresponds to a composition of linear transformations than learning the actual mechanics of doing the computation. You can get good at matrix multiplication without knowing what is going on. I view that as a less valuable activity than learning a…
For understanding the mechanics of matrix multiplication, I found it useful to think in an analogy consisting of a grid of two layers of pipes; One -- the input-pipes -- coming in one direction, and the other -- the output pipes -- laid in an orthogonal direction. Then there would be "taps" in the cross-sections between the input and output pipes, representing the numbers (multiplication factors, really) in the matri…
That's pretty cool. For me, everything changed when I started thinking in terms of column spaces and linear transformations. Matrices are not just a series of multiplications and additions. They describe so much more.
Re: Matrix Multiplication
#114This is a cool example of what Bret Victor calls an "Explorable Explanation" [0]. That said, I feel that it's more important to understand how and why matrix multiplication corresponds to a composition of linear transformations than learning the actual mechanics of doing the computation. You can get good at matrix multiplication without knowing what is going on. I view that as a less valuable activity than learning a…
For understanding the mechanics of matrix multiplication, I found it useful to think in an analogy consisting of a grid of two layers of pipes; One -- the input-pipes -- coming in one direction, and the other -- the output pipes -- laid in an orthogonal direction. Then there would be "taps" in the cross-sections between the input and output pipes, representing the numbers (multiplication factors, really) in the matri…
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Re: Matrix Multiplication
#115Not sure if anyone here has mentioned this yet, but it's useful to think of matrix multiplication as the most natural extension of one-dimensional algebra into multiple dimensions. First, remember that people first wrestled with and found solutions to "1-D" equations like "ax^2 + bx + c = 0". When others came along asking about how to extend these results into multiple dimensions so that previous results might still…
I'm sorry, but that's not how that works.
Your example is of a second-degree equation, which can't be solved by linear algebra. Linear algebra was invented to solve systems of linear equations, and matrices first arose as a convenient notation for expressing those. Matrix multiplication came later and is now understood to correspond to the composition of linear morphisms on vector spaces.
If you try to extend the (very few) methods we have for obtaining exact solutions to polynomial equations in one variable to higher dimension, you (a) fail and (b) invent algebraic geometry.
Re: Matrix Multiplication
#116Matrices are linear maps, and matrix multiplication composes the linear maps. Now statements like the determinant of the product is the product of the determinants, the trace is preserved under similarity transforms (since similarity transforms just rewrite the linear map in a different basis), etc. are intuitive. I've always felt that these explicit calculations don't really get to the point. You can memorize them a…
Maybe you know something I don't, but I don't find these statements about the determinant and trace intuitive just from knowing that matrix multiplication corresponds to composition of linear maps. They can be made intuitive by appealing to geometry, but I don't see how to do that just by staring at the algebraic picture.
Actually, proving them without picking a basis and then showing that what you get is invariant under base change is quite nontrivial and involves throwing a lot of heavy machinery around; see for example Coffman's non-coordinate proof of the statement about the trace in http://users.ipfw.edu/CoffmanA/pdf/book.pdf
For the determinant, if you want to do everything without picking a basis, you're basically proving that the exterior power operation defines a functor on the category of finite-dimensional vector spaces, which isn't that bad to do, but you somehow have to explain what all those words mean along the way.
Re: Matrix Multiplication
#117Earlier quoted context omitted.
Axler's book is very good. An alternative to consider is Paul Halmos's much older Finite-Dimensional Vector Spaces [0]. Both books take the same basic approach, and the proofs of the major theorems are substantially the same. (Halmos's book is well known and well liked and was probably Axler's starting point.) Axler's book covers more ground (most notably, Halmos presents the polar decomposition but not the singular-…
Axler's book was free for me from my university's SpringerLink thingy.
Due to your post, I checked mine -- free pdf version available! Bingo! Thanks for the tip.
Re: Matrix Multiplication
#118This is a cool example of what Bret Victor calls an "Explorable Explanation" [0]. That said, I feel that it's more important to understand how and why matrix multiplication corresponds to a composition of linear transformations than learning the actual mechanics of doing the computation. You can get good at matrix multiplication without knowing what is going on. I view that as a less valuable activity than learning a…
This is why I'm a big fan of Axler's Linear Algebra Done Right . The book's emphasis is on the concepts behind the calculations rather than the calculations. 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 introd…
This is why I'm a big fan of Axler's Linear Algebra Done Right. The book's emphasis is on the concepts behind the calculations rather than the calculations.
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.
Re: Matrix Multiplication
#119Earlier quoted context omitted.
I really like that. And matrix * matrix would just have to be 3D, with a whole cube of taps. Sure, not easy to illustrate, but the concept extends well. It also makes it more obvious what the complexity of the operation is.
Actually the complexity of matrix multiplication is not known. Current best is n^2.37
There is a proven lower bound [1], though.
Re: Matrix Multiplication
#120Earlier quoted context omitted.
This is why I'm a big fan of Axler's Linear Algebra Done Right . The book's emphasis is on the concepts behind the calculations rather than the calculations. 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 introd…
This is why I'm a big fan of Axler's Linear Algebra Done Right. The book's emphasis is on the concepts behind the calculations rather than the calculations. 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…
But surely these sort of meaningless examples with randomized numbers like at the site aren't that useful? I mean, a "real-world" example of matrix multiplication would be having a proper translation or rotation matrix and then visualizing how it maps a point to its image.