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Nowadays we represent the system of $m$ linear equations $$\sum_{i=1}^na_{1i}x_i=y_1$$ $$\sum_{i=1}^na_{2i}x_i=y_2$$ $$\vdots$$ $$\sum_{i=1}^na_{mi}x_i=y_m$$ as $\mathbf{Ax}=\mathbf{y}$, where $(A)_{ij}=a_{ij}$ is an $m\times n$ matrix, $\mathbf{x}$ is an $n\times 1$ column vector, and $\mathbf{y}$ is an $m\times 1$ column vector. Call this the "column picture." But we could just as well have represented it by the transposed equation $$\mathbf{y}^T=\mathbf{x}^T\mathbf{A}^T$$ where we now deal with row vectors rather than column vectors. Call this the "row picture." I have two questions:

(1) Can anyone point to a specific historical reference in which a linear system of equations was represented using the row picture?

(2) Are there any deep reasons for preferring the column picture to the row picture? Or is it fair for us to describe the column picture as totally arbitrary?

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Matrices are really just linear transformations: i.e, they represent functions. In functional notation $f(x)$, the variable (vector) comes after the function (matrix). I don't know if there's more to it than this. –  Robert Mastragostino Feb 1 '13 at 1:33
    
@Robert: yes, right, but I'm asking if anyone could tell an interesting story about how it might amount to more than this. The question is obviously related to why we prefer left to right multiplication in general. –  symplectomorphic Feb 1 '13 at 1:34
    
Because as Robert points out we usually write $f(x)$, so we want that in matrix representation this corresponds to $Ax$, with the matrix on the left side. To do this, once we have agreed that the matrix product is row times column, the only possibility is that $x$ is a column. –  Giuseppe Negro Feb 1 '13 at 1:41
    
Algebraists often prefer their mappings on the right rather than the left. For example, Herstein's "Topics in Algebra" is written that way. –  Robert Israel Feb 1 '13 at 2:14

2 Answers 2

My late professor S. Amitzur, a rather well-known algebraist, wrote a book (in hebrew) in which he systematically writes functions on the right: $\,xf\,\,,\,xA=b\,$ , etc.

When I asked him why would he do such a thing being that the huge majority of books are written the other, "more usual", way, he said: "Most algebraists gave up the algebraic usual notation and adopted the analysts' notation, being the former a more natural and easier to handle notation in algebra. I won't surrender" .

Of course, he said the above in a jokingly mood, yet he consistently used his way to write down stuff in his classes, and this posed some major challenges for greenhorns in linear algebra (and even for graduate students like myself, when trying to adapt things from one writing way to the other one)

For example: the matrix representation of a linear transformation is defined in most books as the transposed coefficients matrix resulting from applying the lin. trans. to some basis and writing the resulting vectors as lin. comb. of some other (or the same, in case of operators) basis. With the algebraists' notation one does NOT take the transpose but directly the resulting matrix, what makes things easier though pretty confussing for someone checking things in other books.

Thus, nowadays, a lin. transf. $\,f:\Bbb R^n\to\Bbb R^m\,$ has a matrix representation of order $\,m\times n\,$ , whereas with Amitzur's notation we get a $\,n\times m\,$ matrix, which is a little simpler, I guess.

And then one has to write "row vectors" to the left of matrices instead of column vectors to the right of matrices. But for that all remains the same.

Another rather interesting example is with products of permutations in the symmetric group, where most authors choose to carry on from right to left, consistently with the usual definition of functions' composition, ,and yet here and there one can still find people who does the other way around, a la algebraists...

Ittay mentions, apparently, Amitzur's book, and I believe him when he says it isn't very popular...now, perhaps, but 20-25 years ago it was very popular, in particular in my school, the Hebrew Unviersity in Jerusalem, and in spite of being rather elementary (1-2 years in linear algebra), I still use it here and there for consultations.

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We prefer the notation $Ax=b$ since it goes in the same direction as reading English does, that's all (much like for functions we write $f(x)$ and not $(x)f$, even though we could have). In languages that are written from right to left for instance some people consider it useful to change the convention. There is a whole linear algebra book written in Hebrew where the mathematical conventions are chosen to mesh better with the Hebrew text. It is not very popular though, to say the least. Usually once adopts the standard mathematica notation even if it created typographical difficulties with the ambient natural language.

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In "Representations and Characters of Groups" by James & Liebeck the notation $(x)f$ is used for applying the function $f$ to $x$. It's rather disconcerting. You might think it would be nice to read composition in the order that the functions are applied till you realize that you've been doing it the other way for so long it takes a noticeable amount of effort to change. –  Jim Feb 1 '13 at 1:50
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As I think Jim's comment suggests, it doesn't seem to me that the fact that English is written left to right, all on its own, can explain why we prefer f(x) to (x)f. For as Jim reminds us, (x)f models composition more 'directly', so that (x)fg means apply f, and then apply g. So I'm not entirely convinced. Is there something more to say? –  symplectomorphic Feb 1 '13 at 1:55
    
yes, but f(x) reads naturally as 'f of x' which is how we usually like to think of functions as processes. –  Ittay Weiss Feb 1 '13 at 1:57
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Let me play devil's advocate here, because I want to make sure we're actually hitting on good reasons. Your reply just seems to push the question back to why we prefer "f of x" to "x in f" or "x through f" or something similar -- all these locutions seem to me to capture equally well the idea of a function as a process (which would seem to just require some kind of indexing of input to output). In other words, both orders display the process equally well. Maybe a more precise answer would be that we prefer active constructions (what f does to x) to passive (what is done to x by f). –  symplectomorphic Feb 1 '13 at 2:12
    
In contexts where the functions themselves are of interest, and we're mostly not writing the argument $x$ (for example, when studying permutations), I much prefer that $fg$ mean "$f$, then $g$", even though it contradicts the usual $f(x)$ convention. –  Ted Feb 1 '13 at 3:30

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