2 ms·
Well, from the historical point of view I think you're right. But in an abstract sense, matrix is a presentation of a linear operator in a specific basis. If w
by dandanua 4y ago
Well, from the historical point of view I think you're right.
But in an abstract sense, matrix is a presentation of a linear operator in a specific basis. If we choose to write it as a transposed matrix, for example, then the matrix product rule would be different (it should correspond to the composition of linear operators).
- nh23423fefe 4y agoBut I think this mixes issues. Vectors are objects from some set (suppose R^n), linear operators some other set (suppose R^n -> R^n). There exists an operation function application which takes an operator and a vector and produces a vector. There is an operation function composition which takes 2 operators to make a third. There is a "functor" from vectors to coordinate vectors, linear operators to matrices, function application to multiplication, function composition to multiplication. The rule for matrix multiplication is constrained by the functor. So you can't choose to write it incorrectly or its not a "homomorphism". This is what I'm trying to express. I don't think you are "free to choose" in some sense. And I wouldn't say the orientation of the vector is caused by the multiplication rule over matrices. I think the "functor" is what is doing the choosing.
- dandanua 4y agoRight, the matrix product rule comes from our choice of the representation of a linear operator (it's what you call a functor). But we are free to choose this functor. I didn't want to complicate things in a comment, the referenced notes have this moment.