4 ms·

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 o

by nivter 3y ago

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 https://www.linearalgebraforprogrammers.com/la/3_mat_vec_mul

eigenket 3y ago
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 3y ago
  > 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 3y ago
  Affine transforms for N dimensions can be represented as matrices/linear maps in dimension N+1
  - eigenket 3y ago
    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 3y ago
      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

by nivter 3y ago

eigenket 3y ago

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 3y ago
> 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 3y ago
Affine transforms for N dimensions can be represented as matrices/linear maps in dimension N+1
- eigenket 3y ago
  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 3y ago
    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