Interesting, I wonder why Facebook has this website blocked, preventing me from posting it on my wall.
I just had the same problem! I wanted to share it with the College group from where I graduated, but I could not do it.
Matrix Multiplication
121–130 of 135 posts
Re: Matrix Multiplication
#122Nice illustration of the "row picture" of matrix multiplication. The "column picture" is equally important.
This intro lecture by Prof. Strang is a really good lesson on that: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...
Also, this short tutorial on linear algebra covers the row + column pictures on the first page: https://minireference.com/static/tutorials/linear_algebra_in... (disclaimer: I'm the author)
Re: Matrix Multiplication
#123How I would explain it: - First, show how to multiply a row vector by a column vector. - Then, show how to "simultaneously" multiply several row vectors by a single column vector by placing the row vectors in a matrix. - Now, similarly, do the same with the column vector. And voila, matrix by matrix multiplication.
im trying to understand this, and what you said appeals to me but i can't quite visualize what you mean. do you have an illustration?
Re: Matrix Multiplication
#124The UI confuses me to the point of frustration.
1. A "Previous" button would help since otherwise you have to go loop through the whole thing to see what actually happened (e.g. how exactly the first step lifts and rotates the matrix, what's getting added to what).
2. The final animation step is a transition from the patently false equation "[matrix1]=[result_matrix]" to the finally correct equation "[matrix1] x [matrix2] = [result_matrix]". This could be avoided if instead of transforming "[matrix1] x [matrix2]" into "[matrix1 plus overlay]=[result_matrix to be filled]" and then doing the 'waterfall', the equation "[matrix1] x [matrix2] = [result_matrix to be filled]" was maintained throughout, perhaps with the moving overlay slightly grayer after lifting [matrix2].
Re: Matrix Multiplication
#125Earlier quoted context omitted.
Is that true for non-square matrices?
Conjugation only makes sense for square matrices. More generally, left and right multiplication correspond to coordinate transformations of the output and input space, respectively.
If A is n x n and P is n x p then P is not square but P^T A P is well-defined.
Re: Matrix Multiplication
#126Awesome. Some concepts in math are so much easier to grasp with an interactive visual. I once made an entire game around matrix multiplication -- such that players didn't even know they learned how to do this. And sometimes, visualizations can help cause discoveries. A lot of progress in theoretical physics came about because physicists 'guessed' what nature's equations should look like...and then got shocked when th…
I'm curious about this game. Is it online?
I built it for the Windows 8 Store (of all places), years ago, in base C++/Cx. All de-published now though. Could dig up a gameplay video if you wish...
Re: Matrix Multiplication
#127It's easier to think of matrix multiplication by computing the matrix elements, which means reducing the problem to NM vector dot products. To get the element of the result matrix at position (n,m), compute the inner product , where r_n is the nth row vector of the left matrix and c_m is the mth column vector of the right matrix. Once you try it, you'll see that it's also much easier to visualize than this strange, u…
Nope, won't ditch the domain. I'd recommend you ditch that type of speech.
Everyone's a critic, don't let it get to you.
Re: Matrix Multiplication
#128Earlier 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 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.
That demonstrates multiplication of a matrix with a vector which is being rotated, but multiplication of two matrices corresponds to composition of the corresponding operators.
This is actually pretty instructive. For instance, rotations in three-dimensional space do not commute in general; hence neither does matrix multiplication.
Re: Matrix Multiplication
#129Earlier quoted context omitted.
I'm curious about this game. Is it online?
I built it for the Windows 8 Store (of all places), years ago, in base C++/Cx. All de-published now though. Could dig up a gameplay video if you wish...
Thanks, but I'm just curious and don't like videos. The world needs more things like that, though -- it was nice to hear of it.
Re: Matrix Multiplication
#130Earlier quoted context omitted.
To back up the parent, unfortunately, the audience is not always qualified to judge the quality of your work - for the simple reason that they have no idea what they are missing! What they are missing from your explanation is "why". Your animation shows how to multiply two matrices, and anyone confused about the mechanics of doing that might find it helpful. But the actual mechanics are not of much use if you don't k…
I don't know how you think that school systems, especially in the west, place more emphasis on "how" than "why". Western schools most definitely place their emphasis on "understanding" rather than "doing". I think that all this does it create a false sense of understanding. Students think they understand why something is done, then when they go to do an actual problem they are totally lost. I think this is a big reas…
>I don't know how you think that school systems, especially in the west, place more emphasis on "how" than "why".
I've taught people at college level, and have seen way too many cases of people knowing how to perform the steps to, say, solve an equation, find a derivative or take an integral without understanding what the operations actually mean, or why bother with all this stuff in the first place.
It's not the students' fault; the concerns I am trying to voice have been articulated by many over the recent years, and are best summed up here: https://www.maa.org/external_archive/devlin/LockhartsLament....