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Comment by dmd | Hacker News Reader

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 this one? I can't think of any other that makes any sense.

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Typically, this visualization isn't taught, in my experience. What's taught is the formula for computing each cell of the result matrix (cell i,j being the dot product of row i of the first matrix with column j of the second). While this is, of course correct, and also the most efficient way to compute the multiplication manually, it's not always clear why the formula is correct.

Hopefully the visualization on the linked page makes this formula's origin clear.

Really? Interesting. We spent some time discussing multiplication by block decompositions in my 100-level linear algebra class -- is this really uncommon?

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.

> 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.

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 or observables in quantum physics, it seem more natural to think of the matrix as acting on the vector, i.e. a point of the geometry or a physical state, not the other way round.

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.

I was taught to go element-by-element: element c_ij of the product = the linear combination a_ik*b_kj. What's presented here is the "same" but represents a higher level of abstraction, so that you think in terms of linear combinations of rows or columns, rather than linear combinations of scalars.

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.

I think this page is meant for someone who has no mental model of matrix multiplication at all and just applies the formula blindly.

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.

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