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I second your request for a detailed explanation of the "technical reason" -- different people kept mentioning it, then saying that they would omit it. I want

I second your request for a detailed explanation of the "technical reason" -- different people kept mentioning it, then saying that they would omit it. I want to know the technical / mathematical details for why, in isolation, row(1, 2, 3) is different from column(1, 2, 3). Like, all the way back down to the purest axioms or geometric intuitions. I understand why the "rules" of matrix multiplication make the distinction -- but what is underneath those rules? What does it mean?

mjburgess 4y ago
By convention, (roughly), row vectors are functions and column vectors are data. [1, 2, 3][2 4 6] Means 1*2 + 2*4 + 6*3
dandanua 4y ago
Matrix multiplication corresponds to the composition of related linear operators. So, it goes down to the representation of a linear operator by a matrix. But we have freedom here. For example, we could choose to represent a linear operator by the transpose matrix (of the usual). But this requires changing the matrix product rule as well (to have the meaning of composition).
sfpotter 4y ago
Say you consider a column vector to be an actual physical vector in space. The geometric way to think of this entity is as a direction (n-1 degrees of freedom) and a magnitude (1 DOF). Because of the definition of matrix multiplication, if we have two column vectors of length n, u and v, and u' denotes the transpose of u, then: u'v = |u| |v| cos \theta where \theta is the angle between the directions determined by the vectors u and v. This means that the matrix u' (a row vector) can be thought of as a linear transformation mapping v to a measurement which encodes the angle between u and v (it's proportional to the scalar projection onto u). For this reason, it makes sense to think of column vectors as "vector vectors" and row vectors as linear transformations, which are not the same thing. Sometimes these linear transformations are called "linear functionals", but this point isn't particularly important. The important distinction is between "geometric objects in space" and "operators which encode actions on that space". You can think of these things as having different types. To actually answer your question: in the above, there is no reason I couldn't have decided that row vectors are "vector vectors", resulting in column vectors being thought of as linear functionals. There is no technical reason motivating this distinction as far as I know. Column vectors are simply the convention in mathematics, while computer graphics did it the other way around for a while.
- shanusmagnus 4y ago
  This is getting close to what I wanted -- thank you. But now you introduce what might be the real heart of it: the definition of matrix multiplication. I know the rules of this operation, and how matrix multiplication distinguishes rows and columns, and the geometric interpretation of a vector multiplied by a matrix. But where does that definition come from in the first place? This might be too ill-posed to answer, it's been years since I was familiar enough with everything to know how to express exactly my confusion. But if you (or anyone) wants to take one more crack, I will appreciate it :)
  - bee_rider 4y ago
    I don't think I'm able to give a real crack at the "where does it come from" question, but an example that might be intuitive -- if you had a system of equations, like high school algebra stuff: a w + b v = y c w + d v = z If you know w and v, and want y and z, this is equivalent to a matrix multiplication: [a b] [v] = [y] [c d] [w] [z] And if you are solving that classic highschool problem "you know y and z, what are v and w," it is equivalent to the linear system solve operation, or multiplication by the inverse of the matrix. [v] = ([a b])^-1 [f] [w] ([c d]) [g] I apologize for the formatting. Imagine any time I've got parens or brackets on multiple lines, they are their big equivalent that envelopes both lines.

mjburgess 4y ago

By convention, (roughly), row vectors are functions and column vectors are data. [1, 2, 3][2 4 6] Means 1*2 + 2*4 + 6*3

dandanua 4y ago

Matrix multiplication corresponds to the composition of related linear operators. So, it goes down to the representation of a linear operator by a matrix. But we have freedom here. For example, we could choose to represent a linear operator by the transpose matrix (of the usual). But this requires changing the matrix product rule as well (to have the meaning of composition).

sfpotter 4y ago

Say you consider a column vector to be an actual physical vector in space. The geometric way to think of this entity is as a direction (n-1 degrees of freedom) and a magnitude (1 DOF). Because of the definition of matrix multiplication, if we have two column vectors of length n, u and v, and u' denotes the transpose of u, then: u'v = |u| |v| cos \theta where \theta is the angle between the directions determined by the vectors u and v. This means that the matrix u' (a row vector) can be thought of as a linear transformation mapping v to a measurement which encodes the angle between u and v (it's proportional to the scalar projection onto u). For this reason, it makes sense to think of column vectors as "vector vectors" and row vectors as linear transformations, which are not the same thing. Sometimes these linear transformations are called "linear functionals", but this point isn't particularly important. The important distinction is between "geometric objects in space" and "operators which encode actions on that space". You can think of these things as having different types. To actually answer your question: in the above, there is no reason I couldn't have decided that row vectors are "vector vectors", resulting in column vectors being thought of as linear functionals. There is no technical reason motivating this distinction as far as I know. Column vectors are simply the convention in mathematics, while computer graphics did it the other way around for a while.

shanusmagnus 4y ago
This is getting close to what I wanted -- thank you. But now you introduce what might be the real heart of it: the definition of matrix multiplication. I know the rules of this operation, and how matrix multiplication distinguishes rows and columns, and the geometric interpretation of a vector multiplied by a matrix. But where does that definition come from in the first place? This might be too ill-posed to answer, it's been years since I was familiar enough with everything to know how to express exactly my confusion. But if you (or anyone) wants to take one more crack, I will appreciate it :)
- bee_rider 4y ago
  I don't think I'm able to give a real crack at the "where does it come from" question, but an example that might be intuitive -- if you had a system of equations, like high school algebra stuff: a w + b v = y c w + d v = z If you know w and v, and want y and z, this is equivalent to a matrix multiplication: [a b] [v] = [y] [c d] [w] [z] And if you are solving that classic highschool problem "you know y and z, what are v and w," it is equivalent to the linear system solve operation, or multiplication by the inverse of the matrix. [v] = ([a b])^-1 [f] [w] ([c d]) [g] I apologize for the formatting. Imagine any time I've got parens or brackets on multiple lines, they are their big equivalent that envelopes both lines.