Notation: is it proper to multiply matrix with a vector represented using an n-tuple $\begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix}(x_1, x_2)$?

Question

I've recently noticed talking to a few classmates from another school in undergrad engineering, they are denoting a vector using n-tuple i.e.

$x = (x_1, x_2, x_3, \ldots, x_n) \in \mathbb{R}^n$

It is fine with me but then I noticed that they would proceed to multiply a matrix (for instance a $2 \times 2$ matrix $A$) as:

$Ax = \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix} x = \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix}(x_1, x_2)$

I was raised to denote $x$ as a column vector, so

$Ax = \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix} x = \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix}\begin{bmatrix}x_1\\ x_2\end{bmatrix}$

feels more comfortable to me. Truthfully, it also feels more correct.

Would $Ax = \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix} x = \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix}(x_1, x_2)$ be considered to be correct as well? Is it good practice?

Answer 1

In at least one course that I've been to the $n$-tuple notation $(x_1,\dots,x_n)$ was used as a convenient way to write column vectors on one line. So in that course our convention was

$$(x_1,\dots,x_n)=\begin{pmatrix}x_1\\ \vdots\\ x_n\end{pmatrix}$$

$$(x_1,\dots,x_n)\neq\begin{pmatrix}x_1& \dots& x_n\end{pmatrix}$$

In other words adding commas turns a row into a column (which is a bit confusing but saves space on the page). In this case multiplying by a matrix on the left would make perfect sense.

Answer 2

As long as what is meant is clear, I don't see any harm. I would not write that for fear of ambiguity however, as I more naturally view $(a,b)$ as a row vector than a column vector, and would assume $A(a,b)$ is undefined when $A$ is 2 by 2.