[dionysus8]

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 transformations is key, like in computer graphics or physics. It’s like watching math and geometry dance together!

[eigenket]

The rest of your comment doesn't really seem specific to geometric algebra. You're just interpreting matrix multiplication as composition of linear maps, which is indeed a very useful and sometimes intuitive perspective.

[nivter]

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

[eigenket]

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.

[ReleaseCandidat]

> 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).

[whiterknight]

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

[eigenket]

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.

[whiterknight]

Agreed. Context: This sub thread is discussing how to interpret matrices as an abstract thing that does something as opposed to the mechanics of multiplying

[bmacho]

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

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

[sva_]

3blue1brown's linear algebra 'course' is excellent on this https://www.3blue1brown.com/topics/linear-algebra