As someone who is studying the subject, I would be greatful if someone can please explain the technical reasons for this. I do not understand why I can't write them as row vectors and do the same Linear Algebra trickery (which we do with column vectors) another way around.
That's not just a mechanical computational consideration. One important interpretation of matrix-matrix multiplication is composition of linear transformations. When multiplying AB, you apply multiply matrix A by each column (vector) of B, and concatenate them to form the columns of the result AB. This corresponds to applying the transformation represented by A to a set of basis vectors. This correspondence is essential to the interpretation of matrix multiplication as a linear transformation, so it's important that it work out without additional manipulation.
Matrix-vector multiplication is more or less the heart of linear algebra, so in general it needs to be an elegant sensible operation.
Matrix multiplication is defined as follows: if A is an m x n matrix and B is an n x p matrix, then the (i, j)th entry of C = AB is:
\sum_{k=1}^p A_{ik} B_{kj}
This definition works whether you think of a column vector as a vector and a row vector as a linear functional or vice versa.
How you interpret "what's happening" (whether you're expanding in a basis or taking dot products with basis vectors) depends on how you interpret the rows and columns of a matrix, regardless of the "orientation".
I assume you know everything I am writing below, but maybe it's a clearer version of what I was trying to say in my other post.
Computationally, the definition of matrix multiplication AB = C is identical to multiplying A by each column of B, and concatenating the results as columns of C. The definition works both ways as you said, but this specific interpretation is one consequence of that definition.
Ax is a linear combination of the columns of A, with the values of x as coefficients. This is a direct translation of plugging the x values into a system of linear equations where the coefficients of each equation in the system form the rows of A:
a11 * x1 + a12 * x2 = y1
a21 * x1 + a22 * x2 = y2
[ a11 a12 ; a21 a22 ] [ x1 ; x2 ] = [ y1 ; y2 ]
You can go on like this, but the overall idea is that if you define "a matrix" such that each row contains the coefficients of one equation in a system of linear equations, then you are setting up row vectors to correspond to transformations (as in, y = a1x1 + a2x2), and then it becomes very ergonomic and natural to represent vectors as column vectors.
In short: if transformations go on the left and vectors go on the right, then transformations are rows and vectors are columns.
I understand why the "rules" of matrix multiplication make the distinction -- but what is underneath those rules? What does it mean?
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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
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.
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 :)
a w + b v = y
c w + d v = z
[a b] [v] = [y]
[c d] [w] [z]
[v] = ([a b])^-1 [f]
[w] ([c d]) [g]
I think it's because column vectors are less of a pain to write on the blackboard.
I'm being serious. There is no reason to say that columns are vectors and rows are linear functionals. The other way around is a perfectly valid convention. But I did find that when I taught linear algebra, column vectors were just easier to write on the board.