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Three interpretations of matrix products

linearalgebraforprogrammers.com

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Re: Three interpretations of matrix products

#11

Post by nivter »

dionysus8 wrote: post #3

I like the fourth interpretation of matrix multiplication from geometric algebra, which brilliantly encapsulates geometric transformations. This approach shifts our focus from just numbers to the geometry of space, revealing how matrices can elegantly describe rotations, reflections, and scaling. It’s a vivid and intuitive perspective that brings matrix operations to life, especially in fields where visualizing these…

Yup, I consider this interpretation as a matrix being a function that takes in objects like a line, a circle or a convex shape and spits out objects like some other line, an ellipse or another convex shape. It is a level of abstraction where you no longer care _how_ matrix multiplication works - you mostly care about what a matrix does to geometric objects. I covered this aspect not in the above article but in a separate one: https://www.linearalgebraforprogrammers.com/la/3_mat_vec_mul

Re: Three interpretations of matrix products

#12

Post by nivter »

adamnemecek wrote: post #2

I really dig the integral transform one. https://ncatlab.org/nlab/show/integral+transform

This was a joyful epiphany for me when I encountered continuous linear systems for the first time. Another corollary of this is when the kernel is δ(x-y), the resulting integral has the value f(x). I like to see it as a continuous version of expressing a vector as a sum of its components.

Re: Three interpretations of matrix products

#13

Post by greatgib »

Honestly I don't find this article to be very clear or understandable about matrix products. The animations are very confusing.

Probably when you already know how to work with matrices you can understand but otherwise I'm quite sure that you would not understand.

Re: Three interpretations of matrix products

#15

Post by eigenket »

nivter wrote: post #11
dionysus8 wrote: post #3

I like the fourth interpretation of matrix multiplication from geometric algebra, which brilliantly encapsulates geometric transformations. This approach shifts our focus from just numbers to the geometry of space, revealing how matrices can elegantly describe rotations, reflections, and scaling. It’s a vivid and intuitive perspective that brings matrix operations to life, especially in fields where visualizing these…

Yup, I consider this interpretation as a matrix being a function that takes in objects like a line, a circle or a convex shape and spits out objects like some other line, an ellipse or another convex shape. It is a level of abstraction where you no longer care _how_ matrix multiplication works - you mostly care about what a matrix does to geometric objects. I covered this aspect not in the above article but in a sepa…

I feel like matrix multiplication/linear maps are kinda the "wrong" things to be considering when you're talking about applying them to lines or shapes in euclidean space.

For example there is no linear map which maps a line (or segment) through the origin to a parallel line (or segment of the same length) that doesn't pass through the origin, even though these are clearly just the same object shifted around a bit.

A much more natural set of operations is (IMO) the affine transformations since then I can move things around as you expect. I find dealing with linear maps of lines or circles or polygons a bit unintuitive.

Re: Three interpretations of matrix products

#16

Post by bmacho »

dionysus8 wrote: post #3

I like the fourth interpretation of matrix multiplication from geometric algebra, which brilliantly encapsulates geometric transformations. This approach shifts our focus from just numbers to the geometry of space, revealing how matrices can elegantly describe rotations, reflections, and scaling. It’s a vivid and intuitive perspective that brings matrix operations to life, especially in fields where visualizing these…

You probably mean "linear algebra". Geometric algebra[0] is a very specific, also vastly different machinery.

[0] : https://en.wikipedia.org/wiki/Geometric_algebra

Re: Three interpretations of matrix products

#17

Post by ReleaseCandidat »

eigenket wrote: post #15
nivter wrote: post #11

Earlier quoted context omitted.

Yup, I consider this interpretation as a matrix being a function that takes in objects like a line, a circle or a convex shape and spits out objects like some other line, an ellipse or another convex shape. It is a level of abstraction where you no longer care _how_ matrix multiplication works - you mostly care about what a matrix does to geometric objects. I covered this aspect not in the above article but in a sepa…

I feel like matrix multiplication/linear maps are kinda the "wrong" things to be considering when you're talking about applying them to lines or shapes in euclidean space. For example there is no linear map which maps a line (or segment) through the origin to a parallel line (or segment of the same length) that doesn't pass through the origin, even though these are clearly just the same object shifted around a bit. A…

> For example there is no linear map which maps a line (or segment) through the origin to a parallel line (or segment of the same length) that doesn't pass through the origin, even though these are clearly just the same object shifted around a bit.

That's why god (well, projective geometrists) made homogeneous coordinates. Without them geometry isn't much fun when using linear algebra (as you have way too many special cases).

Re: Three interpretations of matrix products

#18

Post by whiterknight »

eigenket wrote: post #15
nivter wrote: post #11

Earlier quoted context omitted.

Yup, I consider this interpretation as a matrix being a function that takes in objects like a line, a circle or a convex shape and spits out objects like some other line, an ellipse or another convex shape. It is a level of abstraction where you no longer care _how_ matrix multiplication works - you mostly care about what a matrix does to geometric objects. I covered this aspect not in the above article but in a sepa…

I feel like matrix multiplication/linear maps are kinda the "wrong" things to be considering when you're talking about applying them to lines or shapes in euclidean space. For example there is no linear map which maps a line (or segment) through the origin to a parallel line (or segment of the same length) that doesn't pass through the origin, even though these are clearly just the same object shifted around a bit. A…

Affine transforms for N dimensions can be represented as matrices/linear maps in dimension N+1

Re: Three interpretations of matrix products

#19

Post by blagie »

This is enough detail that:

- If you understand it, it's a nice visualization but you don't learn anything new

- If you don't, you won't understand it.

Going one click in brings up a paywall, with no pricing. You need to give up your email to get a price.

This feel like a not-very-good business model. This would make a lot more sense as either:

- A fully-baked business model, competitive with other paid resources

- An open-source project on github.

To be a fully-baked business model, it would need:

- Enough teaser content to get people hooked and for people to be able to reshare content

- Things to do (e.g. writing Python code), a place to do it (e.g. an online repl, like most other similar systems), and ways to evaluate it for correctness.

- Clear marketing / branding copy (who did it? what's the privacy policy? what's it cost? etc.)

As an open-source thing, it could slot into a community of similar projects which fill those gaps. It has very nice interactives, but it takes a "tell" rather than a "do" approach, which is helpful in context, but isn't adequate for learning by itself.

Re: Three interpretations of matrix products

#20

Post by eigenket »

whiterknight wrote: post #18
eigenket wrote: post #15

Earlier quoted context omitted.

I feel like matrix multiplication/linear maps are kinda the "wrong" things to be considering when you're talking about applying them to lines or shapes in euclidean space. For example there is no linear map which maps a line (or segment) through the origin to a parallel line (or segment of the same length) that doesn't pass through the origin, even though these are clearly just the same object shifted around a bit. A…

Affine transforms for N dimensions can be represented as matrices/linear maps in dimension N+1

Sure, but that isn't what they're doing on that webpage. They're just directly applying linear maps to shapes in R^2.

You can represent a wild variety of things by linear maps on suitably enlarged spaces.

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