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It's common in mathematics to consider vectors as column vectors because the action of some linear operator represented by matrix A on a vector v is just the ma

It's common in mathematics to consider vectors as column vectors because the action of some linear operator represented by matrix A on a vector v is just the matrix product A*v. Mathematical conventions are really dependent on each other. If you change one — a lot of others become awkward. I wrote short notes about conventions in mathematics to understand them better: (pdf) https://github.com/dandanua/little-endian-vs-big-endian-in-quantum-computing/blob/master/txt.pdf https://github.com/dandanua/little-endian-vs-big-endian-in-q...

nh23423fefe 4y ago
The because seems wrong. I would argue vectors are columns because systems of linear equations are sentences which take up a row in a math paper. the matrix is simply the layout and so R^n is column on the other side of the equals. It's arbitrary.
- 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 ago
    But 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 ago
      Right, 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.

nh23423fefe 4y ago

The because seems wrong. I would argue vectors are columns because systems of linear equations are sentences which take up a row in a math paper. the matrix is simply the layout and so R^n is column on the other side of the equals. It's arbitrary.

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 ago
  But 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 ago
    Right, 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.