> Recent mathematical treatments of linear algebra and related fields invariably treat vectors as columns (there are some technical reasons for this)
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.
Matrix-vector multiplication works correctly when the vector is an Nx1 matrix, but if it's a 1xN matrix you need to take its transpose.
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".
> depends on how you interpret the rows and columns of a matrix
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:
Moreover, in a matrix that corresponds to a system of non-redundant linear equations, the columns correspond to a basis for the image of the linear transformation represented by A. This follows immediately from the definition of a basis.
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.
There’s no real reason to think of column vectors as points in space rather than the other way around when you’re doing linear algebra. It’s just a convention. If you wanted, you could think of a row vector as a point in space and transpose everything. In that case, your dual space of linear functionals would consist of column vectors rather than row vectors.
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?
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.
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 :)
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.
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).
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.
IIRC, the argument from the developers on OpenBSD misc mailing lists is that developing on 30 year museum pieces gives them perspective on the difference between right and wrong. It exposes them to more "stuff" than if it were the other way around.
As someone who reads a lot of history, please head over to r/AskHistorians
It has very strong moderation, and low quality answers are deleted. Their papers cite methods and hypothesis which removes a lot of "fiction" from the equation.
Also, they get quoted in mainstream media too. The content quality is out of this world.
I drive a Tata Nexon EV 30kwh battery, 3.5 KW charger which I bought for Rs 1.6 million (16 lakh). The new ones with 40kwh battery, 7 KW charger costs Rs 1.9 million (19 lakh). On road is lesser due to subsidies (not sure if there are any active subsidies).
Tata Tiago is priced much lesser. I would be vary of range as advertised (unlike many other countries). My car gives me practical range of 220 kms and the advertised range is 312 kms. I drive at 115 to 125 WH per km, my wife drives at 135 to 150 WH per km.
I hold a Class B ISP license. We get letters (not emails, physical letters)from Department of Telecommunication asking us to block http://www.example.com
Do they not also ever request blocks on particular IP addresses? A lot of collateral damage happened due to blocking whole ranges from AWS and such when Russia was trying to ban Telegram.
Telecom companies cannot take small subscribers to court in India. You don't pay your bills, they can send legal notices, but not much beyond that. We have ~1B+ subscribers. Even if a small percentage goes into dispute, our judicial system will completely collapse.
Unrelated - I use kagi too! I was watching a supervised learning series on YouTube by Killian Weinberger here [0] and he used an email from kagi as an example of spam.
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.