Consider the $n$-tuple $(x_1,\ldots,x_n)$ with entries in some field $K$.
What is the reason for perceiving this tuple either as a row vector, $$ [x_1,\ldots,x_n]$$or as a column vector $$\left[\begin{array}{c}x_1\\\vdots\\x_n\end{array}\right]$$?
To clarify further: All the answers on this site, that I looked up, like this, this or this one deal with the question, which objects should be thought of as row respectively column vectors – opposed to that is what I'm asking: What's reason for making this distinction in the first place ?
Because the thing is, I could define matrix multiplication, like this $$ \left[\begin{array}{cc} a & b \\ c & d \end{array}\right](x_1,x_2):=(ax_1+bx_2,cx_1+dx_2), $$so I only ever need to deal with "tuple-vectors", not row or column vectors; no need to ever talk about row or column vectors. So the thing is, that everything related to coordinates could be done, in the case of vectors, in row or column terms. So if that is possible, why is nobody doing it like that ?
The only benefit that I see from making the row-column distinction – in contrast to using tuples $(x_1,\ldots,x_n)$ which are neither "row" nor "column"-type – is to be gained on a notational level, so that it is easier to remember, for example, how to do matrix-vector multiplication, by "moving" along the rows of the matrix while "moving" along the column of vector.
But that seems a shallow reason, to put up with distinguishing between "row" and "column"-vectors all the time. I hope there's something deeper than that.
Best Answer
If you just have a tuple of numbers then, as you said, there is no difference between column and row. However, if you want to have nice matrix algebra, you have to distinguish between two.
The reason nice matrix algebra exists is the fact that, any $m \times n$ matrix can be understood as a linear map from $K^n$ to $K^m$. In this formalism, a column vector is a map from $K$ to $K^n$ i.e., a vector; but a row vector is a map from $K^n$ to $K$ i.e., a functional.
If you define the multiplication as you did, you will lose associativity, and that is bad.