Hmm. As someone who has never understood/visualized/been taught matrix multiplication in any other way than how it's shown on the linked page, could someone explain the alternative? I.e., if this is new, what mental model do you currently have? Presumably that one would be new and useful for me. Edit: "Typically this visualization isn't taught" -- that's what I"m asking. What visualization were you taught, if not thi…
> What visualization were you taught, if not this one?
Oftentimes, no visualization or other motiviation is taught at all. Students are simply instructed the steps to follow to produce a correct result.
Hmm. As someone who has never understood/visualized/been taught matrix multiplication in any other way than how it's shown on the linked page, could someone explain the alternative? I.e., if this is new, what mental model do you currently have? Presumably that one would be new and useful for me. Edit: "Typically this visualization isn't taught" -- that's what I"m asking. What visualization were you taught, if not thi…
I was taught by the an other visualization[1]. I thought it was the "standard" method since many textbooks (in Hungary at least) teach it this way. [1] https://upload.wikimedia.org/wikipedia/commons/e/eb/Matrix_m...
Unless I'm missing something, this is precisely the kind of visualization that explains how to compute the result, but not why the computation makes sense. I.e. why is matrix multiplication defined this way at all.
Hmm. As someone who has never understood/visualized/been taught matrix multiplication in any other way than how it's shown on the linked page, could someone explain the alternative? I.e., if this is new, what mental model do you currently have? Presumably that one would be new and useful for me. Edit: "Typically this visualization isn't taught" -- that's what I"m asking. What visualization were you taught, if not thi…
Matrices represent maps (functions), and matrix multiplication is map (function) composition. This is very useful because you can easily see how linear algebra generalises to functional analysis and you can easily get an intuitive feel for many theorems from functional analysis if you convince yourself that infinite-dimensional bases are not that special.
Hmm. As someone who has never understood/visualized/been taught matrix multiplication in any other way than how it's shown on the linked page, could someone explain the alternative? I.e., if this is new, what mental model do you currently have? Presumably that one would be new and useful for me. Edit: "Typically this visualization isn't taught" -- that's what I"m asking. What visualization were you taught, if not thi…
One alternative that I find interesting (not the one I was taught first though) is to visualize matrix multiplication as the sum of the rank 1 matrices obtained by multiplying each column of the left matrix by the corresponding row of the right one.
Hmm. As someone who has never understood/visualized/been taught matrix multiplication in any other way than how it's shown on the linked page, could someone explain the alternative? I.e., if this is new, what mental model do you currently have? Presumably that one would be new and useful for me. Edit: "Typically this visualization isn't taught" -- that's what I"m asking. What visualization were you taught, if not thi…
One alternative that I find interesting (not the one I was taught first though) is to visualize matrix multiplication as the sum of the rank 1 matrices obtained by multiplying each column of the left matrix by the corresponding row of the right one.
I don't know why but it seems to make more sense to me if I think of the multiplication in terms of rotating or transposing and moving the vector or matrix so that the columns or rows line up with the ones they are to be multiplied with, the coloured guides are entirely useless to me.
I was taught by the an other visualization[1]. I thought it was the "standard" method since many textbooks (in Hungary at least) teach it this way. [1] https://upload.wikimedia.org/wikipedia/commons/e/eb/Matrix_m...
Unless I'm missing something, this is precisely the kind of visualization that explains how to compute the result, but not why the computation makes sense. I.e. why is matrix multiplication defined this way at all.
Hmm. As someone who has never understood/visualized/been taught matrix multiplication in any other way than how it's shown on the linked page, could someone explain the alternative? I.e., if this is new, what mental model do you currently have? Presumably that one would be new and useful for me. Edit: "Typically this visualization isn't taught" -- that's what I"m asking. What visualization were you taught, if not thi…
You usually think of the product of a matrix with a vector as the matrix acting on the vector, i.e. every row of the matrix tells you what the corresponding component of the resulting vector is. This turns the thing on its head and describes the result as the vector acting on the matrix. Because you usually use matrices to describe operations, e.g. transformations like translations or rotations in (computer) geometry…
Yes, but if you think of the matrix as a collection of basis vectors, then something like a rotation in R3 becomes crystal clear. Your using the same coordinate but moving in arbitrary (orthogonal) directions rather than just right,up,forward. This makes more sense using scalar*vector operations than using dot products.
What is the point of this blog post? Matrix multiplication is shown like this, and in many other similar ways, in many introductory linear algebra textbooks.