Are subscript variables specific identifiers? i.e. Is a subscript j the same meaning for all variables (Xj, Rj, Nj)

Question

So, I'm going through a linear algebra book to try to break into machine learning, and have come across this:

For products $N_1, ..., N_n$, resources $R_1,...,R_m$ are required.
To produce a unit of product $N_j$, $a_{ij}$ units of resource $R_i$ are needed.

Where $i = 1,...,m$ and $j = 1, ..., n$.

How many units $x_j$ of product $N_j$ should be produced if a total of $b_i$ units of resource $R_i$ are available and no resources are left over.
If we produce $x_1,...,x_n$ units of the corresponding products....

So, my question is this: In the entire context of this question and description, does $m$ and $n$ and $i$ and $j$ refer to the same set of numbers? Or, do they refer to different numbers / sets / contexts depending on the "parent" variable ($x$ in $x_a$).

If this is the case, then I can say that there are a total of $n$ products, and a total of $m$ resources. Is that correct? And, no matter where they are used in the question or solution they always refer to that same value / set of values?

Thank you!

Answer 1

Too long for a comment:

So, my question is this: In the entire context of this question and description, does $m$ and $n$ and $i$ and $j$ refer to the same set of numbers?

These symbols refere to the set $\mathbb N$ of natural numbers.

Or, do they refer to different numbers / sets / contexts depending on the "parent" variable ($x$ in $x_a$).

No idea what you mean.

If this is the case, then I can say that there are a total of $n$ products, and a total of $m$ resources. Is that correct?

Yes. That's exactly what the authors wrote - even if the answer to your previous question were no.

And, no matter where they are used in the question or solution they always refer to that same value / set of values?

"they" is presumably $m$ and $n$. They refer -as the authors wrote- to the numbers of products and resources. As long as nothing else is mentioned you can assume that they are arbitrary. $i$ and $j$ are from the sets $\{1,...,m\}$ and $\{1,...,n\}$ as the authors wrote.