Column or row of a matrix? The question is so simple, but I cannot find the answer.
Is $M_i$ (usually) the $i^{\text{th}}$ column of matrix $M$? Or the $i^{\text{th}}$ row?
Since $M_{ij}$ is the $j^{\text{th}}$ element of the $i^{\text{th}}$ row, I would say $M_i$ is the row.
On the other hand we usually work with column vectors and it is therefore unusual to take a row from a matrix and it would be illogical to have a simple notation for something that is used less often.
If $M_i$ is the $i^{\text{th}}$ row, how would I get the $i^{\text{th}}$ column? Surely not $(M^T)_i$!
 A: To start, it highly depends on the problem and the definitions from said problem to denote exactly what $M_i$ stands for, but if we are to assume it be a $\Bbb R^2$ matrix, then we can be somewhat safe to assume that it means the $i^{th}$ row of that matrix since the convention of denoting the size of a matrix starts with the row, i.e. $m\times n$ with $m$ denoting the amount of rows and $n$ denoting the amount of columns of a matrix.
We commonly see the $M_i$ notation to represent a row in a 2D matrix in the field Computer Science and subjects that stem from it. When we start to explore algorithms and code them, it is much easier to think about, write, and explain, a repeated row operation in terms of notation like $M_i$.
For example (in the perspective of a CS approach), suppose we want to sum all the rows of a matrix in order to get a sum of all elements of a matrix if it was flattened into one horizontal vector (in other words, we wanted to combine all values from all the elements in each column into one row via summation).
Then let’s denote the matrix as $M_{ij}$ with $i$ denoting a row in $M$ and $j$ denoting a column in $M$. Then we can write this as such:
$$\sum^{|M_i|}_{i=0} M_i$$
Lastly, to answer your question about the representation of $M$ if it were to be denoting the columns in $M$, a proper way to do this is to establish that $M_{ij}$ is a matrix with $i$ denoting the rows of $M$ and $j$ represents the columns of $M$. With that, we can then use $M_j$ to denote the $j^{th}$ column of $M$ and provide proper context to why we wanted to do that (in the examples above, if you replace the $i$ with a $j$, then we are doing the same sum with columns instead of rows and get a vertical vector instead).
